The equivalence of some combinatorial matching theorems

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THE EQUIVALENCE OF SOME COMBINATORIAL MATCHING THEOREMS

A thesis Presented to

the Faculty of the Department of Mathematics Adelphi University

In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy

by

Philip F. Reichmeider

April, 1978

Acknowledgement

I thank my advisor, Professor E. Levine, without whose

patience and guidance this work would not have been possible. I am also indebted to my committee Professor M. Beresin Professor D. Lubell

Professor W. Meyer

for their suggestions and encouragement.

Many thanks.

il

Table of Contents

Page

Introduction

1

Graph Theoretic Concepts

9

Konig's Theorem

25

Hall's Theorem

40

Dilworth's Theorem

6h

Menger's Theorem

76

Max-Flow Min-Cut Theorem

93

Some Linear Programming Considerations

12^

Bibliography

141

List of Symbols

153

List of Names

154

Index

155

lil

Notation

The usual notation for sets is adhered to ; however, on occasion a set such as { a , a , ... i 2

, a„} is simply denoted by a a ... a . n 1 2 n

In particular this variation is often used for sets consisting of two elements ; that is, { a,b } is often abbreviated as ab.

On the

occasion that a and b are real numbers the context will distinguish

whether ab represents a set or a product of real numbers. Similarly an ordered n-tuple will be denoted by (a , a , ... , a ); 12

n

but on occasion this ordered n-tuple is designated by a a ... a_. 1 2 n This later notation is primarily used in some instances for designating

ordered pairs.

Also complementation of sets is denoted by the use of a slash; that is, the complementation of a set B relative to a set A is denoted

by A/B.

Then A/B ={xeA|x^B}.

Tn particular if a,b e A then

A/ab is the set of elements of A with elements a and b deleted. Articles and texts are referenced in the thesis by placing

their corresponding index in brackets as [

].

1

Introduction

This work, which is expository in nature, explores a class of

combinatorial results called Mengerian Theorems or Matching Theorems. It is generally recognized that the theorems focused upon are essentially

equivalent. Various proofs showing the equivalence of some of these results are readily accessible in the literature ; however, some are

obscured. The objective here is to origanize the relationships between these combinatorial results into one work and to hopefully to simplify their presentations.

Naturally, limitations had to be imposed on the scope of this work. Some theorems and certain directions of generalization had to be wholly ignored. Also, the results considered here are confined to

finite structures. To be sure, in the intervening years since the

appearance of these results there has appeared a vast literature on

the infinite versions and other generalizations of these theorems. Although these versions are of great interest, the author has with

some regret excluded them from this work. In order to be a bit more specific, one can begin with a

celebrated theorem of Philip Hall which appeared in 1935. This theorem is set theoretic in context and provides the condition under which a finite family of sets has a choice function which is one-to-one ( i.e., the members of the family can be represented by distinct elements of the

union ). This condition, known as the Hall condition, is simply that

2

the union of any k sets of the family contain at least k elements. Upon its appearance, Hall's theorem yielded a result in group theory, namely, the left and right cosets of a subgroup (of finite index) can be "commonly represented" by group elements.

But its

deeper significance came many years later when Hall's theorem with its many extensions became the foundation of what is known as trans­ versal theory.

The great impact of Hall’s theorem in combinatorics is perhaps

in measure due to the simplicity with which it can be stated and applied.

But it is of further interest to note that there are certain

results of a broader and more general nature predating this most celebrated theorem.

One such result, a graph theoretic theorem, was

established by Denes Konig in 1931 [16].

This theorem concerns finite

bipartite graphs and equates the maximum number of edges, no two of

which meet, with the minimum number of vertices which meet every edge of the graph.

When one applies Konig’s theorem to bipartite graphs

satisfying a graph version of Hall’s condition, it yields what is known as the Matching Theorem which is merely a graph theoretic form

of Hall's theorem.

Here the word matching can be viewed in the sense

that the Hall condition assured the existence of a pairing between sets and distinct elements in their union.

Upon translating the word

matching in a more general graph theoretic context, its meaning simply becomes a set of pairwise nonadjacent edges of a graph ( i.e.,

3

the non-adjacent edges match vertices).

About the same time that Konig's theorem appeared there also emerged a theorem due to E. Egervary which vastly generalized a matrix

form of Konig's theorem.

The matrix form of Konig's theorem concerns

the combinatorial properties of matrices comprised of l's and O's.

It

equates the maximum number of 1's, no two of which lie in the same row or column, with the minimum number of rows and columns containing every 1.

As a result of Egervary's contribution it is customary to designate

this matrix version of Konig's theorem as the Konig-Egervary theorem. It should be pointed out that there is a related contribution due

to G. Frobenius in 1917 [75] which provides a special case of the Konig-Egervary theorem.

Apparently, this led to some controversy on

priority as is evidenced by remarks in Konig's work ([16], page 2Uo). Despite the fact that Konig's theorem is broader than Hall's theorem, there is an even earlier and far more reaching theorem due to K. Menger

which appeared in 1927.

Monger's theorem had a topological setting and

it was Konig who recognized its combinatorial and graph theoretic significance.

In one of its simpler forms Menger's theorem equates

the maximum number of pairwise disjoint paths between two distinguished vertices of a graph to the minimum number of other vertices separating

these two distinguished vertices.

Konig originally introduced his theorem

on bipartite graphs as a lemma to initialize an inductive proof of

Menger's theorem.

Another significant development along these lines occurred in 1950

14

with the appearance of a result by R.P. Dilworth pertaining to finite partially ordered sets. Dilworth’s theorem equates the maximum number

of elements, no two of which are comparable, to the minimum cardinality of a partition by chains. The context of Dilworth’s theorem at first

view shows little relationship to either Konig’s theorem or Hall’s theorem. But it was soon recognized that they were essentially equiv­

alent.

[11],[57].

Then in 1956 another important development arose when L.R. Ford Jr. and D.R. Fulkerson introduced their well-known Max-Flow Min-Cut theorem

[37],[38]. This theorem was a result of studies concerning logistic and

economic models. It is indeed a far reaching theorem and ( as will be shown ) encompasses all the previously mentioned combinatorial theorems. Although the Max-Flow Min-Cut theorem can be viewed as

essentially equivalent to Menger’s theorem, Ford and Fulkerson have

provided an elegant constructive proof of their result. These five theorems by Hall, Konig, Dilworth, Menger and Ford

and Fulkerson, are those emphasized in this thesis. A number of proofs of each of these is presented and in two instances the historical proofs are also noted. Although there are many proofs of these theorems in the literature, those selected here are in the author’s view

generally the most elegant available. In particular the author has included the Halmos-Vaughn proof and the Rado proof of Hall’s theorem; Lovasz’s recent and remarkably simple proof of Kenig’s theorem as well as Konig’s original proof; Tverberg’s proof and Perles’ proof of

5

Dilworth’s theorem; Dirac’s proof of Menger’s theorem; and the Ford and Fulkerson proof of the Max-Flow Min-Cut theorem. The primary interest here, however, is not in the proofs of the

classical results themselves but in the relationships that exist between them. Although it is generally recognized and frequently stated that

these theorems are equivalent [57],[41], it is difficult on occasion to locate specific proofs. The interest here is to organize and in some cases simplify the proofs of the equivalence within one work. Here when it is said that one theorem implies another theorem it

not meant in a strict logical sense but in the sense that there is a straight forward route from the first theorem to the second. And when

two theorems are said to be equivalent the route is two way. Since the emphasis of this theorem is in the investigation of

equivalence of certain theorems there will be found throughout the text certain results labeled as theorems which state that the two results are equivalent. These in a strict mathematical sense are not

theorems but are better described as " folk-theorems ". The convention to call these theorems is simply to permit these results to stand out

more prominently in the text.

It is well known that the combinatorial form of the Max-Flow Min-Cut theorem, called the Integrity theorem, is completely equivalent to

Menger’s theorem [11],[3]. It is also readily recognized that the Integrity theorem and/or Menger’s theorem readily imply the results

6

of Hall, Konig and Dilworth [3]. ( Indeed Konig’s theorem is just a special case of Menger's theorem ). Likewise it is well known that that Dilworth’s theorem implies Konig’s theorem and/or Hall’s theorem

[11] and that Hall’s theorem and Konig's theorem are equivalent. However, to assert that all of these combinatorial theorems are equivalent means that one must demonstrate that Konig's theorem and/or

Hall’s theorem ( the bottom echelon theorems ) imply the more extens­ ive theorems of Dilworth, Menger and Ford and Fulkerson. Although such

proofs have been provided, not all are too well documented in the

literature. There are logical paths going from Konig’s theorem and Hall's

theorem to the theorems of Dilworth and Menger. One means of doing

this is offered through some very striking results of A.J. Hoffman

[50]. These results are presented in chapter 3 as extensions of Hall’s theorem. They are then applied in chapter 6 to prove a result

Known as the Circulation theorem which turns out to be completely equivalent to the Integrity theorem ( or the Max-Flow Min-Cut theorem ).

This provides one means of completing the chain of equivalencies. Again in this direction, an elaborate proof of Ford and Fulkerson

which proceeds from Konig's theorem to Dilworth's theorem is presented. But starting with ideas found in [31] it is possible to obtain a very

direct and simple proof going from Konig’s theorem ( in matrix form )

to Dilworth’s theorem. These proofs both appear in chapter 4. Figure 1 below, depicts the lines of logical implications which

7

are pursued in the text. This discussion would not be complete if some noteworthy

developments of the late 19^0’s and the early 1950’s were not mentioned. It was in the late 19^0’s that the field of linear programming evolved

and in particular the duality theorem of linear programming [40] and

the simplex method emerged. ( However, the simplex method has no

immediate bearing on this work ). In

he early 1950’s it was shown

that the duality theorem of linear programming has significant impact

upon the combinatorial theorems discussed here. It was, for example,

used as a principal tool by Hoffman and Kuhn [52] to imply Hall’s

theorem. To make the transition from linear programming to combinatorics, a result of Heller and Tompkins [47] in a form due to Hoffman was

utilized. Chapter 7 is merely intended to provide the flavor of these developments and the results are far from complete. The author has also taken great liberties to modify the original proofs. The reader interested

in more complete details is referred to [60] or to the original sources

[30],[31], [53] which are very readable. This work is confined soley to finite versions of the matching theorems.

In addition the author has almost completely excluded the discussion of matroid theory. There is also an ommision of an early matching theorem due to J. Petersen and a more general theorem due to Tutte [104].

However the author is hopeful that despite these ommisions, there are are sufficient redeeming qualities to warrant a favorable reception of this work.

8

KO 1

ot " 0

which yields p > " 0

is now established; ie, a

o

+ B

o

a + B 0

• Thus the first

equality

= P.

Let C be any set of indenendent edges of G with |c| = B • For each unsaturated vertex v^ choose an edge e^ incident with v^. For

distinct vertices, these choices of e^ will be distinct since there cannot be an edge e joining two unsaturated vertices. These two selected

sets of edges ( 0 and the set of e^ ) cover every vertex of V. Hence B

i

+ (p-2B ) > a 1 "- i

which yields n > ot + B • "ii

To achieve the opposite inequality consider the set of edges D

which cover V with |D| = a . Now if w is an edge of D, v and v cannot both be incident with other edges in D because of the minimum nature

of D. Let v-^,

... , v^ be those vertices of G which are incident with

more than one edge of D. If v^x is an edge of D, then the above implies

2b

that x is not incident with any other edge of D. For each

choose an

incident edge e^ of D. Also let f^, ... , f^ be the edges of D not

incident with any

( i = 1, ... , k ). Then e1, ... , e?, fls ... , f^

form an independent edge set. Thus

k + h < 8

number of edges of D incident with some

. Letting y be the

, p = 2h +y + k .( This

follows from the fact that any vertex other than the v^ comes from either one of the two endpoint

with one of the v. ). But a

of for from a unique edge incident

= |D| = y+ h, hence

n = 2h + y+ k = a

which completes the nroof.I

i

+ ( k+h )< a + B " i i

25

Chanter 2 - Koni^'s Theorem

This chapter presents various proofs of a matching theorem on

bipartite graphs due to D. Konig. The theorem originally appeared as a lemma in Konig's proof of Merger's theorem. In addition a dual

of Konig's theorem and a number of applications are presented. In

later chapters, Konig's theorem will be shown to be equivalent to

many other well-known combinatorial theorems. To illustrate Konig’s theorem, consider the bipartite graph of

figure 2-1. The bold set of vertices provide a minimum sized vertex covering set while the bold edges form a maximum sized set of indepen­

dent edges. In this example a (c) = g (G). Konig's theorem states o i that this equality holds for every bipartite graph. Notice that

a (G) > 8 (G) is trivial since there must be at least one vertex of 0 " 1 a cover on each edge of an independent edge set. Thus the proofg of Konig's theorem are devoted to the reverse inequality.

Figure 2-1 Three proofs of Konig’s theorem are presented below. The first

and earliest is Konig's original proof. The second is an adaptation

26

of Dirac’s proof of Menger's theorem and the third and most recent is due to Lovasz [54]

.

The original proof of Konig1s theorem uses an interesting technique based upon the notion of an " alternating path ”, an idea first employed by Petersen in 1891. To illustrate this concent let G(V,E) be a graph

and let P ~ E be a set of independent edges of the graph G. An alternating

path of G is an open edge path of odd length with edges e , f , ... , i i e such that e , e , ... , e are not members of F while f , f , ... K+l 1 2 ' K+l 12 f^ are members of the independent edge set F;ie, every other edge of

I

the path belongs to F. Again,'any vertex of G incident with an edge of F is called saturated ; otherwise it is called unsaturated. Konig's proof is critically dependent upon the following important observation.

Observation : An alternating path with respect to a maximum sized

independent edge set F cannot be such that both its initial and terminal vertices are unsaturated.

For if it were true that the initial and terminal vertices were

both unsaturated, then the edges of the path not in F along with the edges of F not on the path would comprise an independent edge set

with one more element than F itself which is contradictory. To justify

this, consider figure 2-2 where the edges of F have been labeled as f , i f , ... , fg and the edges of the alternating path have been labeled

e , f , ... , e , f , e. 1

1

&

K

, . Hence f

xv _L

K. t* J.

, ...

, f

p

are the edges of F

not on the path. If the initial and terminal vertices vgand v^

,

27

f k+1

Figure 2-2 are unsaturated, it is claimed that e , e , ... , e , f, 1 2 k+1 k+1

is an independent set with (3+1 edpes. Since

meet

j = k+1, ...

• ••

,

P

( i = 1, ... ,k+l)cannot

, B), no vertex on the path can be saturated

by two ed^es of the independent set. Also the edges e^ for i = 1, ...

,k+l

are mutually independent. For otherwise either vs =v^, contradicting

the openness of the path, or contradicting the independence of f^, ...

, f%.

Theorem 2-1: ( Kbnig ) If G(X,E,Y) is a finite bipartite graph,

then the minimum size of a vertex covering set equals the maximum

size of a set of independent edges; ie, a (G) = B (G). a 1

Proof: ( Kbnig ) Let G(X,E,Y) be a finite bipartite graph and fix some maximum sized independent edge set x Yj» ••• » x^y^where B=B^(G).

Let X and Y be the sets of saturated vertices and X 11 2

and Y 2

be the sets

of unsaturated vertices in X and Y respectively. Notice X^={x^, ... , x^} and Y ={y , ... 1'1

, v }.Classify the vertices of Y into two mutually ' B 1

28

exclusive classes as follows : A vertex y e Y

is called reachable if 1 there is an alternating nath from x to y where x e X . Otherwise y is 2 called unreachable. Define the set C = {c*, ...

, c^of vertices ( which will eventually

be shown to be a cover of size 8) as follows. Let c^ = y^ if y^ is reachable -, otherwise c^ = x^ where i = 1, ... ,R . Obviously C is of

size 8 and

it remains to be shown that C is a vertex cover of G. To

verify this, let ab

e

E(G). Then it will be shown that a E C or b E C,

and for this purpose there are four nossible cases to consider: Case 1. a e X

2

and b e Y . This is impossible. For otherwise ab 2

would constitute an alternating path between unsaturated vertices, and this is in violation of the observation preceeding the theorem.

Case 2. a e X^ and b £ Y^

. In this case ab is an alternating

nath starting in X^ and ending at b. Hence b is reachable and consequently

b E C. Case 3. a

e

X^ and b

e

C. This implies some y

Y^. Suppose a

e

Y

exists such that ay is in the independent edge set and y E C. There is then an alternating nath from a vertex in X

2

to v. The edges ya and ab '

adjoined to this yields an alternating nath between unsaturated vertices.

This again contradicts the above observation; hence a Case l-i, a £ X^ and b £ Y . If a

e

C.

C, then there is an edge ay in

the independent set with y E C. This implies there is an alternating

path through y to which ya and ab can be adjoined. But then b is reachable and hence b e c.

29

From the above four cases the set C is a vertex cover of size 6 (G). Thus a (G) < g (G) and with the already mentioned trivial i o — i

inequality a (G) > 0 (G), the proof is complete. I o — i

In contrast to Konig’s proof, the following two proofs of

Konig’s theorem are more indirect. The first is an adaptation of Dirac’s proof of Menger’s theorem [33] which will be presented later,

while the second is a surprisingly simple proof furnished by Lovasz [54].

Proof: ( Dirac ) Suppose the theorem is false. Then there is a finite bipartite graph G(X,E,Y) with fewest edges for which the theorem

is invalid. The first objective will be to show that any minimum vertex cover of G must be " one-sided

i.e., must be a subset of X or Y.

Suppose G has a cover which is not one-sided. Then there exists a T c X and B C Y both nonempty such that T u B is a minimum sized vertex cover. Consider the induced subgraphs H = G(T,E g (G) implies the contradictory conclusion that ' o ' i Nonin's theorem holds for G. Thus one may finally conclude that any

cover of G is one-sided. If xy E E(G) then L = G-xy is a finite bipartite ^raph for which Konig's theorem is valid. Then by lemma 1-3 either ci (L) =a (G) or o o a (L) =a (G)-l. The first case a (G) =a (L) = B (L)< oo o o 1 =

B (G) contradicts 1

the hypothesis about G. In the second case the removal of xy precludes x and y from occurring in any minimum sized cover of L. Now the minimum cover of L

together with either x or y gives a minimum sized cover of G. But this means that G can have a cover which is not one-sided ( by making the appropriate choice of x or y ) contradicting the conclusion above. Thus no such G can exist and Konig's theorem must follow. *

The more recent nroof of Konig's theorem offered by Lovasz proceeds along the following lines.

Proof: ( Lovasz ) Let G(X,E,Y) be a finite bipartite graph. One

half of the proof is again trivial ; namely To show

a > B

.

ci (G) < B (G), let H be a minimal subgraph of G such that o " i

a (H) =a (G). Once it is shown that the edges in H are mutually indepeno o

dent the nroof is finished since B (G) > B (H) = a (H) = a (G). ' i = 1 o o

31

Suppose that it is not true. Then H has a vertex x

e

X adjacent

to y , y G Y. By the minimal nature of H, a (H-xy,) < a (G) for i = 1,2. 12 o o Hence there is S ç V(G) with |S | = a (G)-l which covers all edges of i i o

H-xy

I

and similarly there is S £ V( G) with IS I =ct (G)-1 which covers ' 2 2 0

all edges of H-xy . Now S cannot cover xy and S cannot cover xy 2 1 1 2 2 without contradicting the nature of a (H). Hence x,v i S and x,y i 0 "11 2

Also note that since xy

and similarly y

2

e

1

is covered by S

2

2 . 2

it must follow that y E S /S 12 1

S /S . 1 2

Let S = S

2 with ISI = t and P = (S /S) w ( S /S)w x. Then 12 12 IR| = 2(a (G)-l-t) +1 = 2(a (G)-t)-l. It can now be shownthat any edge oo . '

of H is such that either it has an endpoint in 2 or both endpoints are in R. For suppose neither endpoint of an edge of H is in 2. This is

certainly true of xy^ and xy? for which it has been observed x,y^,y^ E R.

Any other edge not naving an endnoint in S must clearly have one endpoint in 2 /S ( since S is a cover of H-xy ) and the other in S /S 11 ' I 2 ( since S is a cover of H-xy 2 2

); ie, both endnoints are in R.

Letting T be the smaller of Xn R and Y a

R yields | T | < [

|R| ]=

(G)-t-l. The vertex set To S covers H since T covers every edge

induced by R while the only other edges in H are those having an endpoint

in S. But |T o S| =|t| + |s| = a (G)-t-1 + t = a (G)-l, a contradiction. o o

Thus the edges of H are mutually independent; which completes the proof.I

Before proceeding notice that the condition that G be bipartite

32

in Konig's theorem is essential. Generally if a ^raph is not bipartite

the condition a (G) = B (G) may or may not hold. Indeed the graphs H o 1

and K in figure 2-3 are not bipartite and a (H) = 3 / 2 = B(H) in the o i first case while a (K) = 2 = B (K) in the second.

H :

Figure 2-3

By examining G in figure 2-4 another characteristic of bipartite graphs is illustrated : namelv, a

i

(G) = B

o

(G). That is, the minimum size

of an edge covering set of G ( the bold edges ) equals the maximum size of a set of independent vertices of G ( bold vertices ). This is the dual of Konig's theorem and is an immediate consequence of it. In the dual of Konig's theorem it is assumed that the bipartite graph has no

isolated vertices to assure that a

Figure 2-4

exists.

33

Theorem 2-2: If G(X,E,Y) is a finite bipartite graph without isolated vertices, then the maximum size of a set of independent vertices of G equals the minimum size of an edge covering set of G; ie, p,

0

= a . 1

Proof : From Gallai’s theorem ( theorem 1-4 ) « and from Konig’s theorem ( theorem 2-1 ) a

o

= P

i

0

+ g

= n = a + B 11

0

. Thus g

o

= a

i

.I

The reader might note that Gallai’s theorem shows that Konig's theorem and its dual are completely equivalent.

The study of bipartite graphs has some interesting implications

relating to combinatorial properties of matrices. This results from a natural correspondence between bipartite graphs and matrices of 1's

and O's. If G(X,F,Y) is a finite bipartite graph with x^ e X for 1 < i < m and y, e Y for •• - "=

3

1 < j < n, then A = ( a^^ ), the m by n =

incidence matrix for G,ns defined so that a.. = 1 if x-y.€ iJ 1 J

E and

a. , = 0 otherwise. Conversely, given the incidence matrix it is ij easy to reconstruct the bipartite graph. The graph G and the matrix A

in figure 2-5 illustrate the correspondence described above.

X 1 1

X

i

*y

fl 0 0 0 1 0 10 0

G: x

0 10 0

x

I0 1 1 1J

Figure 2-5

34

In what follows a line of a matrix is either a row or a column of a matrix. A line of a matrix covers the entries on it. A set of lines cover A if every nonzero entry of A lies on at least one of the lines.

Nonzero entries of A are independent if and only if no two lie on a common line.

The relevance of these concepts lie in the fact that a vertex covering set in G corresponds to the notion of a line covering set in

A. In addition a set of independent edges in G corresponds to a set

of independent elements of A. Thus a maximum sized set of independent

edges of G corresponds to a maximum set of independent !'s in A. From Konig's theorem the following result ( generalized by Egervary [34]

)

is now apparent.

Theorem 2-3: ( Konig-Egervary ) Let A be a m by n matrix of 1's

and O's. The minimum number of lines covering A equals the maximum number of independent I's of A.

The proof of the Konig-Egervary theorem, as indicated, follows

from the interpretation of vertex coverings and edge independence for bipartite granhs in terms of line coverings and independence of nonzero

elements for incidence matrices. In fact, proceeding in reverse, it becomes apparent that Konig's theorem is completely equivalent to the

Kbnig-Egervary theorem. To illustrate the above discussion, the set of bold vertices of G

35

in figure 2-5 is a minimum sized cover and corresponds to rows 1 and 4 and column 2 which is a minimum line cover of A. In addition the set of hold ed^es of G is a. maximum set of indenendent edges and corresponds

to the hold !'s of A which is a maximum set of independent I's of A. The Konig-Egervary theorem can he used to produce an earlier but

closely related result due to Frobenius on the expansions of a deter­

minant . To introduce

Frobenius’ theorem consider figure 2-6 which

illustrates the expansion of a 3 by 3 determinant of 1’s and O's. Notice that each term in the expansion is zero and that there is a 2 by 2 zero submatrix of A. The relation between a zero submatrix of a square matrix

and its determinantal expansion was first pointed out by Frobenius [75]

10 1 0 10

110 - 101 + 000 - 000 + 101 - 101

0 10

Figure 2-6

Theorem 2-4 : ( Frobenius ) All the terms in the det erminant al expansion of an n by n matrix A are zero if and only if A

contains an r by (n-r+1) zero submatrix where I < r < n.

Proof: Suppose every term in the determinantal expansion is zero.

In this case the maximum number

of independent nonzero elements of

A must be less than n. Then by the Konig-Egervary theorem the minimum

number ct

of lines which cover every nonzero element of A is less than

.

36

n. Hence some 1 rows and a -1 columns cover every nonzero element of A. o

This implies that the remaining n-i by n-o. +i submatrix is a zero o submatrix with (n-i) + (n-a +i) = 2n- a > n. This is the desired result. o o

Conversely suppose there exists an r, 1 < r < n, such that some r by n-r+1 submatrix of zeros of A exists. Clearly the nonzero elements are contained in the remaining n-r rows and r-1 columns implying that

n

< (n-r) + (r-1 ) = n-1. By the Konig-Egervary theorem this implies o —

< n-1 or 3 < n. Thus there is no set of n independent non-zero 1 — i elements of A. Hence every term in the determinantal expansion of A

3

vanishes as reouired.I

Another application of the Konig-Egervary theorem yields a result due to G. Birkhoff which Pertains to doubly stochastic matrices. This

result requires the notion of a permutation matrix which is a matrix T T . P of I's and O's such that PP =1 where P is the transpose matrix and

I is an identity matrix. If P is a square permutation matrix then P

has a single 1 in each row and each column. Figure 2-7 illustrates a square permutation matrix of order 3.

Figure 2-7

37

As an aside, permutation matrices are useful in the study of graph

isomorphisms. For example if G(X,E,Y) is a bipartite graph with a corresponding incidence matrix A, then premultiplication or post­

multiplication of A by a permutation matrix corresponds to a relabeling of X(G) or Y(G) respectively. Figure 2-8 illustrates this point. Namely,

A is the incidence matrix of G while PA is the incidence matrix of H

*111

0 10

x2

0 0 1

P = 1

0

0

GIO X

yl

x

y2 corresponds

X

Figure 2-8

which is merely a relabeled version of G ( that is X is relabeled ). Thus permutation matrices are revelant in investigating graph invariants under relabeling ( ie, under isomorphisms ) via the incidence matrix.

A stochastic matrix is a matrix of nonnegative reals such that the sum of every row is 1. A square matrix A is doubly stochastic

if and only if A is a matrix of nonnegative reals with every line sum

38

equal to 1. Matrix A in figure 2-9 is a doubly stochastic matrix. As illustrated, it can be represented as a linear combination of

permutation matrices where the coefficients are nonnegative and sum

to 1. Such a linear combination is referred to as a convex linear combination.

Z3/8

0

5/8

0

1/2

0

0^

/ 1

0

0

o'1

^0

0

1

0^

1/2

0

1

o

0

0

0

0

1

A =

+1/2

= 3/8 0

zo

0

1

o'

0

1

0

0

0

0

0

1

0

0

°)

+1/8

1/2

o

1/2

0

0

0

1

°

1

0

0

3/8

0

0 \

n

1

0/

V1

0

0

0

Figure 2-0 Figure 2-9 illustrates a theorem proved by 0.Birkhoff in 19^6

[621

that any doubly stochastic matrix can be presented as a convex

linear combination of permutation matrices. A more general result

provided by the next theorem relies unon the Konig-Egervary theorem.

Theorem 2-9 : Let A be an n-th order matrix with nonnegative entries having a line sum M > 0. Then there are positive reals and permutation k k matrices , ( i = 1, ... , k ) such that A = E c.P. and E c. =M. 1=1 1 1=1 1

Proof : The n-th order matrix A with constant line sum M must

contain n inderendent nonzero elements. Otherwise the Konig-Egervary

39

theorem implies fewer than n lines cover A. Let such a cover have e rows and f columns where e + f < n. Then (e+f)M is greater than

or equal to the sum of all the elements of A, hence (e+f)M > nM or e + f > n, a contradiction. Now consider n independent nonzero elements of A, say b , ... Let c^ = min( b , ...

, b^) and B= A- c^P^where

, b^.

is the n-th order

nermutatioq/natrix with l ' s in positions corresponding to the nonzero elements b^, ...

, b^. Then B is an n-th order matrix having

constant line sum M-c with at least one more zero element than A. 1

Repeating the argument until the line sum is reduced to zero yields k k A = £ c.P. and M = £ c. . I i=l 1 i=l ïfhen the matrix A is doubly stochastic theorem 2-5 becomes :

Coro11ary 2-6:( C.Pirkhoff ) Let A be an n-th order doubly stochastic matrix. Then there are positive reals c. and n-th order permutation k k matrices P. where i = 1, ... ,k such that A = £ c.P. and £ c.= 1. 1 i=l 1 1 i=l 1

If in theorem 2-5 the matrix A has nonnegative integer entries , then it is clear from the proof that the quantities c^ can be selected

as nonnegative integers. This leads to the following.

Corollary 2-7: ( Kdnig ) Let A be a square matrix of 11s and O's such that every line of A has exactly k entries which are l's.

Then A is a sum of k permutation matrices.

1*0

Chapter 3 - Hall's Theorem

In 1935 Philip Hall presented and proved a theorem concerning

the representation of a family of sets. It soon became apparent that

Hall's theorem could be viewed as an instance of Konig's theorem in

which a minimum vertex cover of a bipartite graph is a one-sided cover.

This view of Hall's theorem is not to minimize its contribution. Indeed this theorem in its particular setting was the catalyst to stimulate

much activity in what can be classified as transversal theory. The main objective of this chapter is to demonstrate this strong relationship between Konig's theorem and Hall's theorem. In addition, two of the many available proofs of Hall's theorem are presented. The

first is due to Halmos and Vaughn and the second is due to Rado. A few variations of Hall's theorem are also presented, particularly those

useful in establishing the connection to Konig's theorem. To this end certain relationships between families, bipartite graphs and incidence matrices are pointed out. Hall's theorem is also applied to develop

conditions for the existence of a common system of representatives of a pair of families. Finally, weighted versions of Hall's theorem are

presented and yield results useful in chapter 6. No attempt is made to be complete since L. Mirsky1s Transversal

Theory [21] and L. Mirsky and H. Perfect's " Systems of Representatives " [57] are available for this purpose.

41

Let A = {Ai J i

e

1} "be a family of nonempty sets with S =^Ai.

A system of representatives of the family G , denoted "by SR, is a family of elements { x.

| i e I } of S such that

E A^ for i E I.

( Thus an SR is simnly a choise function. ). A system of distinct representatives of the family '(, denoted by SDR, is a system of

representatives i

| i e l j such that i f J implies x^ f x^.

A set of elements in an SDR is called a transversal of U . In general,

a family may have no SDR, one SDR or more than one SDR. The conditions for a family to have an SDR is the focal point of this chapter. Let Q = { A^ | i e I } and for , J = I, let A

= U A. . ieJ Then Q satisfies the Hall condition : if and only if | Ay | >_ | J | for every J £ I. It is a trivial observation that if a family t( has an SDR, then the Hall condition is satified since there are already the

]J^members of the SDR in each Ay. Hall’s theorem states that in case of a finite indexing set, the Hall condition is also sufficient for an SDR to exist.

The two proofs of Hall's theorem presented below are confined to

finite families of finite sets but can easily be extended to all finite

families. Relaxing the hypothesis to encompass finite families of arb­

itrary sets is easy and left to the reader. However, relaxing the hypothesis to include infinite families is not possible without impos­ ing further restrictions. To see this consider the infinite family

A

0

={x,x,x, ... } and 12 3

A. = { x. } for i = 1,2,3, .... This i i

family certainly satisfies the Hall condition but clearly has no SDR.

42

Hall’s "theorem is applicable to arbitrary families of finite sets, but the details are not included here.

Theorem 3-1 : ( P. Hall ) Let J = {

| i

e

1} be a finite family

of finite sets. Then the family U has a FDH if and only if \ satisfies the Hall condition | A^ | > |J| for all J Ç I.

Proof : ( Halmos-Vaughn ) If 6! has an SDR, the Hall condition is satified since for any J ~ I

from the SDR in

there are already |J| distinct elements

A^. .

The proof of sufficiency proceeds by induction on 111

. The basis

for induction, |l| = 1, is obvious. Suppose that the theorem holds for |I| = n-1. To demonstrate its validity for [I| = n consider the follow­

ing two cases. ( In these cases, I = { 1,2, ...

,n } ).

Case 1. For every J Ç I, J | I suppose that |A a e A^and consider

> |j|. Choose

= A^/a for all i E I’ = 1/1. Then for any

J s I',

B. = (A./a) = ( (J A. )/a. Thus |B j = |A /a| icJ 1 ieJ isJ J _> |Aj| - 1 > | J | - 1 or |Bj| | JI and by the induction hypothesis

there is an SDR

a E A

1

{b^|

is 11 } of the family

is distinct from each b., i E I’. Hence i ’

{

| i E I'} . But

{ a,b , ... , b } form 2 n

an SDR of Q . Case 2. Suppose there exists a J such that ]j|
|B U Al — 1 J

and Av- = (A U A

ieK

).

= A^ .

Is I and Ik I >,B v A^I — 2 K

= IT I

Then

such that

where AT = U

A. i

Consider S U T = ( B U AT) U ( B u Ay- ) = 1 J = %

By the hypothesis this family satisfies the Hall condition,

U

so that ]S u T| > 1 + |J v K[. Tn addition S n T = ( B u A ) n 1 J ( B u A% ) T- Aj

A^ z Aj^

But the family indexed by J

K satisfies

the Hall condition so that | S n T|_> | J r\ K| . Together these results yield |J|+|k|>|s| + |t|= |SvT|+|sr\ t| >1 +|Ju K| + |Jn K| = 1 +|j|+|K|,

a contradiction. Thus either

{ B , A , ... I

2

, A_} or {B , A , ... n

2

2

, A } n

satisfies the Hall condition.

By repeating this process, one element at a time cam be removed from the sets

{Aj | i e 1} arriving at a family of singleton sets

satisfying the Hall condition. Such a family must consist of distinct elements and is thus an SDR of Q = { A^ | i e I } .

Diverging briefly, it is interesting to note that Rado's proof

of Hall's theorem resides in a more general setting of matroids [25].

A matroid

defined on a set S is a non-empty collection of subsets

of S satisfying the following properties :

* if A e • if A,C E

and B £ A then B E

and |A, + 1 = |C|then there is an x E C/A

such that A U x €

.

The concept of a matroid is a generalization of the concept of

independence in a linear vector space. An infinite set of vectors is independent whenever every finite subset of these is also independent. Analogously a matroid

on S is a finite character matroid if the

following additional property is satisfied: • if A is a subset of S such that every finite subset of A belongs

to "hj then A itself belongs to

.

H5

The members of

i are called independent sets while all other

subsets of S are dependent sets. A rank function p is defined on the subsets A of S with respect to its matroid h] in the following manner:

If for each positive integer n there exists an independent subset of A with at least n elements then

p(A) = m . Otherwise

p(A) = k where

k is the largest number of elements in an independent subset of A. Rado's proof of Hall's theorem can be applied verbatim to prove

the following central theorem of matraids.

Theorem 3-2 : het

be a matroid with rank function p on a set

of elements S. The finite family Q ={ A.

| i e l } of subsets

possesses an independent transversal if and only if for each

J Ç I yields

p(Aj) _>

|j|

.

Returning, a family does not satisfy the Hall condition when it

has no SDR. In this case every SR of such a family has at least one repeated element. To broaden the scope of Hall's theorem, it is of

interest to introduce a notion of a partial transversal. A partial

transversal of a family Ci is a transversal of a subfamily of G

.

Hall's theorem can easily be strengthened to yield necessary and

sufficient conditions for a family to possess a partial transversal

of a prescribed finite cardinality. For suppose a family Cl is such

that |Aj| _> |J| - o ( the minimum 6

is called the deficiency of the

familyQ). Adjoining 6 new elements to each set of the family then

46

assures an SDR of the new family. But then at least r = |I| - 5 elements of the transversal are elements of S and comprise a partial

transversal of the family C . Hence :

Theorem 3-3: Let Q = {A^ J i E I } be a finite family of subsets

of a set S and let r be a positive integer with r
of size k coincides

elements of

with P or P . . Choose some u E P and v E P . such that v < u. Let max min max min —

69

Ck = uv and P* = P/Ck

Then Ck is a chain and | P* | < | P| implies P’

contains a maximum sized antichain of k-1 elements. Then by the induction k k-1 which hypothesis P' = U where are chains. Hence P = P' u uv = u i=l i=l is the desired result. I

The extent of the equivalence between Dilworth’s theorem and the

previously presented matching theorems are now considered. It will be

seen that Dilworth’s theorem directly implies both Konig's theorem and Hail’s theorem. In the reverse direction one seems to require a more

elaborate line of reasoning.

Theorem 4-2: Dilworth’s theorem implies Konig's theorem.

Proof: Let G(X,E,Y) be a finite bipartite graph. Then G can be viewed

as a finite poset such that for x E X and y

e

Y let x £y if and only if

xy e E(G). By Dilworth's theorem the minimum size chain cover equals the

maximum size antichain of < V(G), £ > . But any minimum size chain cover of < V(G), < > corresponds to a minimum size edge cover of G(X,E,Y) and a

maximum size antichain of < V(G), £ > corresponds to a maximum size set

of independent vertices in G(X,E,Y). Consequently Dilworth’s theorem 1 one has a

* the dual of Konig's theorem. Then by theorem 1-4

- B

implies a

o

0

= g

i

, Konig’s theorem. I

Theorem 4-3: Dilworth's theorem implies Hall's theorem.

TO

Proof: ( Mirsky and Perfect ) Let ij= { Ai I 1 < i < n } be a family of subsets of a set ( aj | l£j£m} and suppose the Hall condition

is satisfied. Consider a partial ordering among A^’s and a^'s such that aj < Aa if and only if a^ £ At . If the set of k elements of aj’s and

h subsets of Ai's, a^, ...

, aj , A^, ... , A^ , are incomparable then

none of the a. , ... , a. belong to the A^ , ... , A^ and by the Hall J1 Jk 1 h condition h < m-k. Thus any antichain has length at most m. Then the length of the maximum antichain is m since the a^'s are incomparable.

By Dilworth's theorem the cardinality of the minimum chain cover is also m. With a possible relabeling let the minimum chain cover be a^A^, a^,

... , a^, at+1, ... , am, A^, - -,

which implies t = n and a^ ... ,

an comprise an SDR of Q . I

As indicated the reverse implication to those above is not nearly so simple. It will now be shown that Dilworth's theorem is a consequence

of Konig’s theorem and hence indirectly of Hall's theorem also. Two proofs will be presented. The first proof is an approach due to Ford and Fulkerson.

In this line of reasoning the following two lemmas are first established. Given any finite poset < P, £ > a corresponding bipartite graph is constructed.

Let G(P,E,P) be a bipartite graph the left and right hand sides of which are disjoint copies of P. Then let xy £ E(G) for x,y £ P if and only if x < y in < P, . Then

71

Lemma 4-4: If I is the cardinality of a set of independent edges in G(P,E,P) excluding edges tt where t e P then there are J chains

covering < P, < > such that I + J = | P |.

Lemma 4-5: Let K be the minimum size of a vertex cover in G(P,E,P). Then there is an antichain of length L in < P, £ > such that

K + L = | P | .

Before proving these lemmas it is shown how they are used to imply

Dilworth’s theorem.

Theorem 4-6: Konig’s theorem implies Dilworth’s theorem.

Proof : Applying lemma 4-4 let ; be the maximum number of independent edges in G(P,E,P) and J be the corresponding size of a chain cover of
such that Î + J = | P |. Then by lemma 4-5 I + J = | P| = K + L

where K is the size of a minimum vertex cover in G(P,E,P). Then by Konig's theorem I = K implying the size of a chain cover of < P, £ > J equals the size of an antichain L. Thus the minimum size of a chain cover C and the

maximum size of an antichain of < P, £ > A are related by C £ J = L £ A. Hence C < A. But trivially C > A. Then C = A the desired result. :

To complete the Ford-Fulkerson argument, the following proofs of

lemma 4-4 and lemma 4-5 are provided.

72

Proof of lemma 4-4: Let G(P,E,P) be relabeled G(X,E,Y). Any t e P

is unsaturated in X or Y if it is not incident with any of the I independent edges. A broken path starts

at an unsaturated element of X whose odd

edges are vertical ( tt e G(P,E,P) ) and even edges belong to the independent

set of I edges. This path continues until it is forced to terminate at an unsaturated vertex of Y. No vertex is on more than one broken path or the

independence of the I edges is contradicted. Let J be the number of broken

paths in G(P,E,P). Each such path corresponds to a chain in < P, . There are J unsaturated vertices in P each initiating a broken path with

J ■ | P | - I or J + I = | P | . R

Proof of lemma 4-5: Let G(P,E,P) be relabeled G(X,E,Y). If p - x^ £ X

is an element of a minimum size vertex cover of G then p = y^ £ Y cannot be. For if it were then there are edges x^y^ and x^yj such that x% and yj are not in the cover. But by the transitivity of < P, < >,

implies xk < x

xk £ xi £ xj which

so that x^yj £ E(G). But neither x% nor yj is in the minimum

sized vertex cover, a contradiction. Three types of vertices of G(X,E,Y) are distinguishable ; elements of the cover in X, elements of the cover in Y

and elements in X, say x^, such that neither x^ nor y^ where p = xi = y^ belong to the cover. This last set of vertices comprise an antichain in P. Thus if K is the number of elements in a minimum vertex cover and L is the

number of elements in the antichain then K + L = | P |. B

73

There is even a more direct route from Konig’s theorem to Dilworth’s

[31] . To see this, let < P,£ > be

theorem based upon ideas found in

a poset with P =

{ 1,2, ...

. Then < P,£ > can be represented by an

,n }

n-th order incidence matrix of 1’s and O’s A = ( a^j ) such that a^j = 1

if and only if i £J ; otherwise a^

= 0. Now any set of independent l’s

of A can be viewed to correspond to

a disjoint chain cover of < P,£ >

To see this let

.

of a disjoint chain cover k < >. Then since the chains of the cover are

be a chain j,

... < j.


p. Thus the length of some antichain is less than or equal to the minimum number of disjoint chains covering < P, £ > . But

trivially q < p so that p=q

which is the desired result. fl

The chapter concludes with a dual of Dilworth's theorem. This theorem which is of lesser interest and more readilÿ. established than Dilworth’s

theorem is none the less appropriately included. With this purpose in mind one begins with the following observation. Since an antichain can contain at most one element of a chain it is apparent that the minimum number of

antichains covering < P, < > is at least as large as the maximum length of a chain of < P, < > . Theorem 4-6 below which is the dual of Dilworth's

theorem asserts that this inequality is indeed an equality. In figure 4-1 no fewer than three antichains cover the poset, namely, { e, cd , ba),

while the maximum length of any chain is 3 such as edb.

75

Theorem 4-7 : Let < P, < > be a finite poset. Then the minimum size antichain covering of < P, £ > is equal to the maximum length

of a chain of < P, £ > .

Proof : The proof proceeds by induction on the maximum length of a

chain of < P, £ > . The basis of the induction is trivial. Suppose the theorem is valid on posets whose maximum length is less than n. Let


be any poset having n as the maximum length of a chain.

Furthermore, let M be the set of maximal elements of < P, £ > . Clearly M is an antichain and any maximum length chain meets M in one and only one

element. Hence the maximum length chain in

P/M is n-1 and by induction

P/M is covered by n-1 antichains. Hence < P, £ > can be covered by n anti?

chains. 8

76

Chapter 5 -

Menger1s Theorem

Thus far the theorems of Konig, Hall and Dilworth have been

presented and relationship between them explored. The next result to be considered, due to K. Menger in 1927, predates these theorems and is of a vastly more general nature. Indeed Konig's theorem was introduced

as a lemma which served to prove Menger's theorem by induction. Thus, as will be seen, Konig’s theorem is simply a special case of Menger’s

theorem. Menger’s theorem was originally developed in a topological setting but it is the graph theoretic form which is of interest here.

Let u and w be nonadjacent vertices in a simple graph G. A uw vertex path in G is a path having u and w as the initial and terminal vertices.

A collection of uw vertex paths which are pairwise disjoint except at

u and w is called independent ( or uw independent ). A collection 5 of vertices not containing u or w is called a uw vertex separating set if every uw path contains a vertex of S. Thus the removal of a uw vertex

separating set leaves u and w in different components of the resulting

graph. It is immediate that the minimum cardinality of a uw vertex separ­ ating set must be greater than or equal to the maximum number of uw independent paths. Monger’s theorem asserts that these two quantities

are equal. The following elegant proof of Monger’s theorem is due to Dirac [33]

.

77

Theorem 5-1 : ( Menger ) If u and w are nonadjacent vertices of a simple graph, then the maximum number of uw independent vertex

paths equals the minimum cardinality of a uw vertex separating

set.

Proof: ( Dirac ) Let m be the minimum cardinality of a uw vertex separating set and M be the maximum number of uw independent paths. Then, as noted above, m

M. It will be shown by contradiction that

there are m uw independent paths. Suppose Menger*s theorem is false. Let G(V,E) be a simple graph

with fewest edges for which the theorem fails. Hence there are non­

adj acent vertices u and w for which there are m separating vertices and fewer than m vertex independent paths. Now, there is no vertex s of V in G(V,E) such that usw is a path.

For if there were, Menger's theorem would hold in G/s implying that there are m-l uw separating vertices and m-l uw independent vertex paths. Upon reconstruction of G there are m uw independent vertex

paths which is a contradiction.

Let S C V be a uw vertex separating set of minimum cardinality m. Let Pu be the set of all vertex paths between u and exactly one vertex in S and Pw be the set of all vertex paths between w and exactly one

vertex in S. The paths of Pu and Pw have only vertices of S in common. For otherwise the existence of a uw path in G not meeting 2 is implied,

78

contradicting S as a uw separating set. In addition either all the

vertices of S are adjacent to u or all the vertices of S are adjacent

to w. For otherwise one arrives at a contradiction in the following manner. Consider the graph formed by adjoining the edges us where s e S to the paths Pw and the graph formed by adjoining the edges sw where

s e S to the paths of P . Both of these are simple graphs with fewer edges than G. Consequently both graphs have m uw independent vertex paths. But joining the Pw and Pu parts of these paths at elements of

S produce m uw independent vertex paths in G. This is a contradiction. Now remove an edge e of G not meeting either u or w. On this graph G/e Menger’s theorem is valid. Indeed since M < m there are at most m-l

vertices separating u and w. Any uw path of G not meeting these uw separating vertices of G/e must meet this removed edge. Since m is

the minimum cardinality of a uw vertex separating set of G, the exist­ ence of such a uw vertex path is assured. Thus either vertex of the

removed edge can be used with these m-l separating vertices to separate u and w in G. This contradicts the fact that a minimum uw separating set in G is adjacent to either u or w. Thus Menger’s theorem holds. I

It will become apparent from the discussion that follows that there are many forms of Menger’s theorem. The versions considered here are

those which shed light on the relationship between Menger’s theorem and

other combinatorial matching theorems. The next result is one of the original versions found in Menger’s paper

[56J

and in Konig’s book [16]

.

79

To introduce this, let U and W be disjoint nonempty vertex sets of V(G). Then a set S of vertices, not necessarily disjoint from U

or

W, such that every vertex path between U and W contains a

vertex of S is called a weak UW vertex separating set. Thus U or W is a weak UW vertex separating set. Now a set of vertex paths between U and W are called strongly independent whenever they are

pairwise disjoint; that is, two such paths haven't even vertices of U or W in common.

Theorem 5-2 : ( Menger ) Let U and W be nonempty, disjoint

subsets of V(G). Then the maximum number of strongly

independent UW vertex paths equals the minimum cardinality of a UW vertex separating set.

Proof: Enlarge G by adjoining a vertex u with edges from u

to every vertex of U and a vertex w with edges from w to every vertex in W. Clearly u and w are nonadjacent in the enlarged graph. Applying theorem 5-1 yields the desired result. I

A nearly literal translation of Konig's original proof of

theorem 5-2 is now presented. Although many refinements of Konig's proof can be found, the details presented here are consistent with

the original. The interested reader will find a closer translation in Robacker’s paper.[ 60 1.

80

Konig’s proof is presented here for two reasons. First, it demonstrates the extent to which Menger’s theorem is a generaliz­

ation of Konig’s theorem. Second, it is of historical interest to

note that the technique uses induction on each portion of a ” pulled apart " graph to imply the conclusion. This technique, which is

preceivable in the proofs of Perles, Tverberg and Dirac, can be

traced back to Kdnig.

Proof: ( Konig ) It can be assumed that G(V,E) can not be

separated by fewer than n vertices while every proper subgraph can.

Two cases can be discussed:

Case 1. Let V = U u W. Removing from G all edges between elements of U and all edges between elements of W yields a bipartite graph. Theorem 2-1 implies there are n separating vertices and n independent edge paths between U and W.

Case 2.

o

Let s

eV exist such that s o

i U u W. This case

proceeds by induction on n, the cardinality of the minimum

separating set. The basis of induction is trivial. Now remove from G ali edges adjacent to s . By the choice of G, U and W can o

be separated by r < n vertices, say s*, ... , sr. Since

S = { s , s , ... , s o i r

} is a UW vertex separating set in G, then

r + 1 > n. With r < n, this implies r = n-1. Thus S = { s , s , ... ,s — oi U~-L

where s

$ U v W.

}

81

Partition S such that s. ES 1

Q

if and only if s. | Uu W;

s. E S_ if and only if s. E U; and s. ES

1

if and only if s., E W.

A path of type 1 is a path of G from U/Su to one and only one

vertex of S u o

2 . All such type 1 paths form a subgraph P of G. w u

Similarly a path of type 2 is any path of G from W/S* to one and

only one vertex of S^v

Su. All such type 2 paths form a subgraph

Pw of G. Any vertex s E V(PU) r\ V(Pv) is an element of 2. For otherwise the existence of a UW path not meeting S is implied; a contradiction.

In addition E(PU) n E(Pw) = $. For otherwise both vertices

of any common edge would belong to S. This is impossible by the definition of a type 1 and a type 2 path.

To each vertex sj E S there is a UW path of G with only this vertex of 2 on it. For otherwise n-l vertices of 2 distinct from

s^ separate U and W. Thus s@ is the only vertex of S on some UW vertex path. Thus one edge adjacent to s° belongs to Pu and

another belongs to Pw. Hence neither Pu nor Pw are null graphs and both have fewer edges than G. Let n^, n^, n^ be the cardinality of 2^ Su, Sw respectively.

In Pu, U/Su and S^v Sw can not be separated by fewer than n^+ n^

vertices of some separating set. For otherwise this separating set together with Su would constitute fewer than n vertices separating

U and W. Any UW path of G not meeting Su initiates from U/Su and

82

meets some

e

v

Sw. This is a path of type 1 which must meet

some element of a minimum set separating U/Su and S@ u Sw. Such a

minimum separating set has cardinality n° + n^. Since Pu has fewer vertices than G, by the induction hypothesis there are n

o

+ ru W

vertex independent paths between U/Su and S^ Sw. Each such path

meets precisely one vertex of S^v S^. Since Su is disjoint from the vertices of P^, no vertex of Su belongs to these paths. Let

those paths meeting Sw be denoted by T1( x ... , T n ywhile those paths meeting S be denoted by U', ... , U* . i nQ

By exactly the same, argument as above, there are n^ + nu paths of Pw> say

U", ... , U" 1 no meet S

o

, ... , W^, each meeting one vertex of Su and

each meeting one vertex of S . Joining the paths that o

from Pu and P , n w o

vertex independent paths between U and W

are formed: namely, U , ... , U_ . Thus there are n = n 1 “O o u w

+ n

+ m.

UW independent paths U , ... , U_ , W , ... , W_ , T , ... , T„ 1

no

1

“u

1

“W

of G. This proves Menger’s theorem. I

If G is a bipartite graph, then it is clear that Konig’s theorem is a special case of Menger’s theorem; i.e.,

Theorem 5-3: Menger’s theorem implies Konig’s theorem.

The next form of Menger’s theorem to be presented is closely

related to theorem 5-2 and it is commonly found in modern texts ( see Hararay [14]). To contrast it to theorem 5-2, the following

83

definitions are introduced. Two vertex sets U,W of V(G) are nonadjacent whenever u and w are nonadjacent where u £ U and

w e W. Now if U and W are nonadjacent vertex sets of V(G) then

a UW vertex separating set is strong if it contains no vertices

of U or W and a set of UW independent vertex paths is weak if they are pairwise disjoint in V/U v W ( i.e., a pair of such paths

could meet in U or W ). Then

Theorem 5-4 : Let U and W be nonempty, nonadjacent subsets

of V(G). Then the maximum number of weakly independent UW vertex paths equals the minimum cardinality of a strong

UW separating set.

Proof: Contract all vertices of U to a vertex u and all

vertices of W to a vertex w. The fact that U and W are nonadjacent implies that u and w are nonadjacent in the contracted graph.

Then applying theorem 5-1 yields the desired result. I

Observe that theorem 5—1 is a special case of theorem 5—3

when U and W are singletons. In the above theorems, separability has been measured in

terms of the cardinality of a vertex set and connectivity has

been measured in terms of a number of vertex paths. But there

84

are also useful edge versions of Menger's theorem, referred to as

duals

, where separability is measured in terms of a edge

set and connectivity is measured in terms of the number of independent edge paths. The distinction between these concepts

is illustrated in figure 5-1 where v

is a minimum uw vertex

separating set implying the existence of at most one uw independent

Figure 5-1 vertex path. However the three bold edges comprise a minimum uw

edge separating set and consequently it will imply that there is a

maximum of three independent uw edge paths. The dual of Menger’s theorem was not uncovered until 1956

when a number of independent proofs appeared. Ford and Fulkerson demonstrated it as a consequence of the integrity theorem [ ll ].

The result was also discovered by Elias, Feinstein and Shannon ( 35]

The various edge versions of Menger’s theorem result from

the application of the vertex versions to what is known as the

line graph of a graph. The line graph of G, denoted by L(G), is

85

the graph whose vertices are the edges of G and two vertices of L(G) are adjacent if and only if they are adjacent edges in G. An illustration of a graph and its corresponding line graph is

given in figure 5-2. G:

e

2

Figure 5-2 minimum edge separating set in G

Observe that an

corresponds to a v^v^minimum vertex separating set in L(G); specificly, e , e 2

v

2

, v 3

31

separating e 5

separating v

and v

15

and e

in G corresponds to

in L(G). Also a maximum set of

independent edge paths connecting e* and e^ in G corresponds to a maximum set of independent vertex paths connecting v% and

v

5

in L(G); specificly, e 3

and e e 2

4

connecting e 1

corresponds to v^v^v^ and v^v^v^v^ connecting v

and e 5

in G

and v$ in L(G).

In general, any separating set of edges in G corresponds to a separating set of vertices in L(G) and any set of independent

edge paths in G corresponds to a set of independent vertex paths in L(G). Then corresponding to theorem 5-2 and theorem 5-4 there are respectively:

86

Theorem 5-5 : Let P and Q be nonempty, disjoint subsets of E(G). Then the maximum number of strongly independent PQ

edge paths equals the minimum cardinality of a weak PQ edge separating set.

Theorem 5-6: Let P and Q be nonempty, nonadjacent subsets of E(G). Then the maximum number of weakly independent PQ edge

paths equals the minimum cardinality of a strong PQ edge

separating set.

Although it is not germane to the present discussion, it is

of interest to note that there exist various mixed forms of Menger’s

theorem [ 14 ] . Looking ahead, the equivalence between the integrity theorem

and Menger’s theorem ( chapter 6 ) requires the establishment of the validity of Menger’s theorem to a wider variety of graphs ( in particular, multigraphs, directed graphs and directed multi­

graphs ). A multigraph M(V,E) is a finite nonempty collection of

vertices V along with a family of edges E. The restriction that a

unique edge join a pair of distinct vertices is relaxed. Two or more edges joining a pair of distinct vertices are called multiple

edges. ( One can generalize even further by allowing edges to Join

87

a vertex with itself. Such edges are called loops. However,

this discussion will be confined to multigraphs without loops. )

All the aforementioned forms of Menger's theorem are valid for multigraphs. While the vertex forms do hold, they

will not be discussed here. It is the edge separating form of

Menger's theorem which is of primary interest in what follows. To see its validity for multigraphs the notion of a first sub­ division of a multigraph is introduced. The first subdivision

of a multigraph is obtained by introducing a dummy vertex on each edge of a multigraph. Thus the vertex set of G is V v E and two vertices are adjacent if and only if one corresponds to a vertex v of V and the other corresponds to an edge of E

incident with v. ( See figure 5-3. )

First subdivision of G

Figure 5-3 Now to prove the edge version of Menger's theorem for

multigraphs suppose P and Q are disjoint edge sets of a multi­

graph M(V,E). Let P' and Q’ be edge sets of the first subdiv­ ision corresponding to P and Q respectively; that is, if

uw=eeP(orQ)inM then ue, ew

e

P* ( or Q* ) in the

first subdivision of M. Let ev, where e E E(M) and v E V(M),

88

be an edge of a minimum weak P'Q’ separating set in the first

subdivision of M. This clearly implies that e is an edge of a weak PQ separating set in M(V,E). Furthermore, the strong independence of a collection of PQ edge paths in M is unaffected by the introduction of dummy vertices in the first subdivision

of M. Thus there is a preservation of the cardinality of a

minimum weak PQ edge separating set and the cardinality of a

maximum set of PQ strongly independent paths upon first sub­ division of M(V,E). Hence theorem 5-5 easily extends to multi­ graphs .

Theorem 5-7: Let P and Q be nonempty, disjoint subsets of E(M). Then the maximum number of strongly independent

PQ edge paths equals the minimum cardinality of a weak PQ edge separating set.

There is yet a wider applicability of Menger’s theorem to what are known as directed graphs or directed multigraphs.

These forms of Menger’s theorem are presented primarily because of their usefulness in implying the integrity theorem of net­ work theory. In addition, theorem 5-8 below, turns out to be

a convenient tool to imply all the other forms of Menger’s theorem. Basically

a directed graph is a simple graph in which

89

every edge is given a direction. In the context of directed graphs the vertices will be referred to as nodes and the

edges will be referred to as arcs. More precisely, a directed graph, denoted by D(N,A), consists of a finite nonempty set N

( whose elements are called nodes ) and a set A of ordered

pairs of distinct nodes. An ordered pair belonging to A is called an arc. ( Again the reader is reminded that the ordered pair (u,w) will be represented by uw ). Figure 5-^

illustrates the concept of a directed graph. The ordering of the arc uw

is pictorially denoted by placing an arrow on

the arc pointing toward w.

Figure 5-4 A collection of distinct nodes u - v , 0

of N forms a uw

... , v

11

= w

node path whenever v^jV^ for i = 1, ... , n

are arcs of A. A collection of uw node paths are independent

if they are pairwise disjoint except at u and w. Furthermore, a uw node separating set 8 is a set of nodes not containing

either u or w such that any uw node path contains at least one node from this set. ( Here, however, the removal of these

90

nodes from the graph does not necessarily leave u and w in

different components of the resulting graph ).

By a nearly identical proof to that of Dirac’s proof the directed counterpart of theorem 5-1 becomes

Theorem 5-8: Let u and w be nodes of a directed graph

such that uw is not an arc. Then the maximum number of independent uw node paths equals the minimum cardinality

of a uw node separating set.

If each edge of a simple graph is replaced by a pair of

oppositely oriented arcs, then every set of independent uw ( or wu ) node paths corresponds to a set of independent uw vertex paths in the simple graph and every uw

( or wu ) node

separating set corresponds to a uw vertex separating set in the simple graph. Thus the maximum cardinality of a set of

independent paths and the minimum cardinality of a separating

set are preserved by such an alteration. Thus theorem 5-8, the directed node form of Menger1s theorem, clearly implies

theorem 5-1, the undirected vertex form of Menger’s theorem.

In this manner, theorem 5-8 can be used to imply all forms of Monger’s theorem considered here.

It turns out that the directed counterparts of theorem 5-6

and theorem 5-7, applied to what is termed a directed multi­

91

graph, is of interest. Indeed as will be seen in chapter 6,

theorem 5-10 below is completely equivalent to the integrity theorem of Ford and Fulkerson.

Towards extending Monger's theorem in this manner there

is the following. A directed multigraph

M(N,A) consists of

a node set N and a ( finite ) family A of arcs. The distinguish­ ing feature here is that a directed multigraph allows for many

arcs joining nodes u and w of N. Then if P and Q are disjoint are sets of a directed graph or a directed multigraph, a PQ arc path v v , v v , ... , v r 0112 n—1 n

,v

is a set of adjacent arcs

such that v v E P and v ,v E Q. A collection of PQ are o 1 n-± n

paths are strongly independent

if and only if they are pair­

wise arc disjoint. On the other hand a collection of PQ arc paths are weakly independent if and only if they are pairwise are disjoint up to P and Q ( that is, they may have arcs of

P and Q in common ). Also an arc set is called a PQ are separating set

whenever any PQ are path contains at least one of these arcs. A PQ are separating set which is disjoint from P and Q is

called strong

while a PQ arc separating set which includes

at least one arc of P or Q is called weak. Then the directed

counterparts of theorem 5-6 and theorem 5-7 are respectively

92

Theorem 5-9: Let P and Q be nonempty, disjoint arc sets

of a directed multigraph. Then the maximum number of strongly independent PQ are paths equals the minimum cardinality of a weak PQ are separating set.

and

Theorem 5-10: Let P and Q be nonempty arc sets of a

directed multigraph such that no arc of P is adjacent to an arc of Q oriented in the same direction. Then the

maximum number of weakly independent PQ are paths equals the minimum cardinality of a strong PQ arc separating set.

( Note: The hypothesis requires that if p E P and q E Q are arcs then one cannot have _

P

.)

93

Chapter 6- Max-flow Min-Cut Theorem

In this chapter a central theorem of network theory is considered. This theorem, the max-flow min-cut theorem, and its combinatorial

counterpart, the integrity theorem, are due to Ford and Fulkerson. The . latter can easily be used as a principle tool to imply the theorems of Konig, Hall, Dilworth or Menger. But it is also the case that Hall's

theorem ( in weighted form ) can be used to imply the integrity theorem.

The purpose of this chapter is to explore the rich relationships that

exist among these theorems. There is no attempt to provide a complete exposition of network

theory or of its combinatorial consequences. A more complete exposition of the subject can be found in Flows and Networks by Ford and Fulkerson.

The theorems presented here concern directed graphs but in this context a directed graph is called a network. Then a network or a directed network is a directed graph B(N,A) whose vertex set N and

edge set A are referred to as a node set and an arc set respectively.

In addition it is convenient to distinguish two nodes of the network, one of which be commonly labeled s and the other t. The node s will be called the source and the node t will be called the sink. This chapter

is limited to considering finite networks and a typical illustration of such a network is given in figure 6-1.

There is a need to consider various functions from the arc set A to the reals. Let h denote such a function ; that is, h: A

R. If

94

s

Figure 6-1

a = (u,w) E A then h(a) or h(u,w) will denote a. typical value of the

function. If U and W are subsets of N, then h(U,W) = E h(u,w) where the summation is over u E U, w E W and (u,w) e A ; ie , h(U,W) is the sum

over those arcs of A directed from U to W. If U = {u} then h(U,W) is

conveniently noted as h(u,W). Likewise if W = {w} then h(U,W) is noted

as h(U,w) . A straightforward set theoretic result for subsets U,W,Z of N yields

If U

h(UUW, Z)

= h(U,Z) + h(W,Z) - h(Un W,Z) and

h(Z, UU W)

= h( Z ,U) + h( Z ,tf) - h(Z,Un W).

W = (|) then clearly

h(UU W, Z) The output of

= h(U,Z) + h( W,Z) and h(Z, U UW) =h(Z,U) +

h(Z,W)

the function h at node v of N is defined as h(v,N)

and likewise the input of the function h at v of N is h ( N, v ). By a proof similar to the handshake lemma ( lemma 1-1 ) one has the following :

Lemma 6-1: If h is any real valued function defined on the arcs of a finite network, then E h(v,N) = E h(N,v). veN veN

Of special interest are nonnegative real functions f on A, called

95

flows, which have the property that the input at v E N equals the out­

put at v where v

s,t; ie , f(N,v) = f(v,N)

( v f s,t ) .

Then the net flow out of s, denoted by f ( s), is defined as f(s) = f(s,M) - f(N,s).

Likewise the net flow out of t, denoted by f(t), is defined as f(t) = f(t,M) - f(N,t).

Then,

Theorem 6-2: Let f be any flow function defined on A of a finite network D(îI,A). Then f ( s) =

( The quantity

-f ( t) .

-f(t) can be viewed as the net flow into t ).

Proof: From lemma 6-1

X f(v , M) = VEM

L f(N,v) or VEN

Z (f(v,N) - f(N,v)) + f(t,Il) - f ( N, t ) = 0 ,t Since f(v,M)=f(N,v) for any flow where v s,t, f(s,N) - f ( N, s ) +

veNjV^s

f(s,N) - f(N,s) + f(t,N) - f(N,s) = f(s) + f(t) = 0 hence f(s)= -f(t).l

For any flow function f, the quantity f(s) is commonly called the

value of the flow and is denoted by u. A flow function, or any function defined on A of D(N,A), is conveniently displayed on the pictorial representation of the network by placing the value of the function

adjacent to the arc. This is illustrated in figure 6-2 where the value

96

of the flow can readily seen to be \> = f(s) = 10.

* v

Figure 6-2

In addition to the flow function there is another related function called the capacity function, denoted by c. For present the capacity function is considered to be a nonnerative real function. The capacity

function assigns to each a E A a value c(a) and generally flows are constrained such that f(a) _< c(a). Intuitively a flow can be thought

of as an amount of a commodity that can be transferred from the initial to the terminal node of an are per unit of time when the

network is in steady state. The capacity can than be thought of as

a physical constraint on the maximum amount of flow through the arc.

The capacity of a network can conveniently be displayed in the illus­ trations by placing its value to the right of the value of flow as

depicted in figure 6-3. o o

v

v

8 o

2'5

10,15

V

5,5 2

4 4

Figure 6-3 Any arc on which the flow is zero is said to have zero flow and

any arc on which the flow equals the capacity is said to be saturated.

There always exists a trivial flow in any finite network ( namely,

$7

f(a) = 0 for a £ A ) and with a finite arc capacity on the network, a standard compactness arguement implies the existence of a maximum flow value through the network. By a maximum flow function one means a

function for which V is a maximum. A maximum flow function is not necessarily unique.

To investigate the max-flow min-cut theorem, it is necessary to

focus on subsets of A which separate the source s from the sink t. As in the previous chapter, if C is a subset of arcs such that every s-t

edge path ( or in this context s-t are path ) includes some arc of C, then C is called a separating set of arcs. Now there are certain

separating sets called cuts which are of special interest. Namely, let X Ç N be such that s

e

X and t £ X; let X = N/X ; and let (X,X) be the

set of arcs (x,y) such that x E X and y E X. Then (X,X) is called a

cut and it is easy to see that a cut is a separating set, ( Note: Any minimal separating set is a cut ). Also,the cut capacity of (X,X) is

by definition simply c(X,X). ( A cut for which c(X,X) is a minimum is called a minimum cut ).

Now the notion of a network is extended to encompass the following: • a directed graph D(N,A); • two distinguished nodes s ( source ) and t ( sink ); * a capacity function c;

• a flow function defined such that f(a) £ c(a) for all a £ A;

However, considerable modification to this initial description of a

98

network will be made as the discussion

progresses.

By considering a network as a model of steady state flow as

previously discussed it becomes intuitively clear that the

value of any flow must be less than or equal to the value of any cut. ( To see this, if the arcs in a cut are”pinched" then all flow would

cease ). This inequality is indeed the case in general as is shown by

the next result.

Theorem 6-3 : Tn any finite network let f be a flow function of

value v and let (X,X) be an arc cut separating s from t. Then

v = f(X,X) - f(X,X) < c(X,X).

Proof: Since (X,X) is a cut, s E X. Then

v = f(s,N) - f(N,s)

and 0 = f(v,N) - f(N,v) for v E X with v f s. Summing over all nodes

of X

v = f(X,N) - f(N,X)

= f(X,X W X) - f(xux,x) = f(X,X) + f(X,X) - f(X,X) - f(X,X) = f(x,x) - f(X,X) Since f(X,X) < c(X,X) and f(X,X)

0 then v = f(X,X) - f(X,X) 0 ( these are called type (ii) arcs ). Then z E N is called reachable from s if and only if there is a chain of pairs from s to z,

sav, sv , v v , ... , v z 1

112

n

such that s R v , v

R v , ...

112

, v R z. An n

arc path from s to z consisting- of arcs of types( i) or( ii) is called an sz augmented path . These concepts are useful in the following constructive proof of the max-flow min-cut theorem;

Theorem 6-h: ( The Max-Flow Min-Cut Theorem of Ford and Fulkerson) In any finite network, the maximum flow equals the capacity of a minimum cut.

Proof: Let flow function f have maximum flow v on the finite net­ work D(N,A). Let X be all those nodes of N which are reachable from s.

Then s E X and t E X. For if this were not so, then t E X and there is

100

an augmented path in D, sv^, v1v2 » ••• » vn^‘ getting

£ > 0 be the

minimum of c(a) - f(a) on all type (i) arcs and of f(a) on all type(ii) arcs it is easy to see that on arcs of type (i) on the augmented path the flow can be increased by £ while on arcs of type (ii) on the aug­

mented path the flow can be decreased by £ .Then the net effect would be to increase the value of the flow from s to t along this augmented

path by an amount £ . But this contradicts the fact that the value of the flow v is maximum. Thus it follows that t £ X as asserted. Then

(X,X) is a cut such that every uw e(X,X) is saturated. For otherwise u e X. Therefore V = f(X,X) - f(X,X) = c(X,X). Finally as observed above,

since V is less than or equal to the capacity of any other cut, c(X,X) is a. minimum cut .8

In order to establish the combinatorial counterpart of theorem 6-4, the hypothesis is confined to integer valued capacity functions on D(N,A).

This in turn permits the strengthening of the conclusion of theorem 6-4 to assert the existence of a maximum flow function which is integer

valued.

Theorem 6-5: ( The Integrity Theorem of Ford and Fulkerson ) If the capacity function for a finite network D(N,A) is integral,

then the maximum flow equals the capacity of a minimum cut. In addition, there is a maximum flow function which is integer valued.

Proof: Suppose the flow function f is an integer flow having

101

maximum value

on the finite network D(M,A). Let X be all those nodes

of N which are reachable fron^. Clearly s £ X and t E X. For if t E X there exists an augmented path in D, sv , v v i

12

,

, v t. Now it is n

easy to see that on an arc of type (_i) the flow may be increased by 1

and on arcs of type (ii) the flow may be decreased by 1. The net effect would be to increase the value of the flow by 1. But this contradicts the assumption that the flow function f is a maximum integer flow. Thus

it follows that t E X

as asserted. Then ( X,X) is a cut and every

(u,w) £ ( X,X) is saturated. For otherwise u E X. Hence w = f(X,X) - f(X,X)

= c(X,X). Finally since the flow value V is less than or equal to the capacity of any other cut, c(X,X) is again a minimum cut.I

The integrity theorem is easily seen to imply the max-flow min-cut theorem. In order to see this, first suppose that the capacity function

e is a nonnegative rational valued function defined on D(N,A). By using

an appropriate integer factor, the capacities on all arcs can be scaled up to integer values. The integrity theorem implies the maximum flow

equal the minimum cut capacity on such a network. The resulting flow function can then be scaled back down to yield the max-flow min-cut

theorem on the given network having a nonnegative rational capacity function. From here, a continuity argument yields the Max-Flow Min-Cut

theorem. Hence

Theorem 6-6: The integrity theorem implies the max-flow min-cut theorem.

102

It is interesting to note that the proofs of theorems 6-5 and 6-6 are constructive. That is, the maximum flow and the minimum cut on

the network D may be found by constructing s-t augmented paths and

sequentially increasing the flow until no further s-t augmented paths exist. During any step, the flow can be increased on a type (i) and

decreased on a type (ii) arc of an s-t augmented path by an amount

equal to the minimum of e(a) - f(a) over type (i) arcs and f(a) over

type (ii) arcs. The algorithm terminates by identifying a maximal

flow and a minimum cut in D. To illustrate the procedure consider the network displayed in figure 6-4. Here an s-t augmented path

can easily be identified as (s,v ),(v ,v ),(v ,v ) and ( v ,t). 4

2

2

4

3

3

V

10 t

S V

V

2

4

Figure 6-4 All but (v ,v ) are type (i) arcs and the minimum of c(a) - f(a) on 4

2

type (i) and f(a) on type (ii) arcs is 3. Hence increasing the flow by 3 units on type (i) and decreasing the flow by 3 units on type (ii)

arcs yields a maximum flow of 13 units as shown in figure 6-5.

v

V

;

9,7

5.5

\

5,2

v

Figure 6-5 Continuing the process yields no further s-t augmented path but does

103

partition the nodes into those reachable from s and those not. Those reachable from s are V = {s,v ,v ,v } yielding a minimum cut (V,V) 1 2 4"

=

{(v ,V ) ,(v ,V )} . 13

4

3

Although a network is conveniently thought to have a single source

and a single sink, multiple sources and sinks offer no obstacle. A network with multiple sources s^ and multiple sinks tj can easily be modified,

without compromising results, to a single source-sink network. This is accomplished by introducing a dummy source s' and a dummy sink t’ with arcs and corresponding capacities (s',s^ with c(s') = c(s^,N) and

(tj,f) with c(tj,t') = c(N„t^) respectively. Figure 6-6 illustrates

this procedure. S I 1

s* 2

s

t

2

t" 2

Figure 6-6 The flow functions thus far have been bounded below by zero and

above by the capacity, that is 0 d(a). The generalization of

theorem 6-5 becomes :

Theorem 6-8 : If an integer flow function exists on a network with

d( a) _< f ( a) _< c( a) for all a e A where d and c are integer valued

functions, then the maximum flow equals the minimum of c(X,X)-d(X,X) taken over all cuts (X,X) of D. In addition, there is a maximum

flow function which is integer valued.

The lower capacity funtion d will be displayed on a pictorial re­

presentation of the network by recording its value to the left of the

corresponding flow. Then in figure 6-7, d(v ,v ) = 3, f(v ,v ) =7 and 2

2

1

1

c(v ,v ) = 9. To illustrate the construction of a maximum flow con. 2 i

strained by both upper and lower capacity, consider figure 6-7. The construction of two s-t augmented paths ( sv , v v , v v Cv t followed '

2

2

4

4

3

3

105

s \ 3,5,5

\

Figure 6-7

by sv , v v , v v , v t ) yields a flow whose value v = 13 is maximum 1 ’

i

u '

u

3

3

and a cut (X,X) where X = { s, v^ ,v^} whose capacity c(X,X) = 13 is a minimum. The resulting network is displayed in figure 6-8.

s v 3,3,5

V

Figure 6-8

The next extension of the max-flow min-cut theorem is a corollary of theorem 6-8 and will be used further on to provide a proof of

Dilworth's theorem. Theorem 6-8 equates the maximum flow v in any net­

work D(N,A) with the minimum of c(X,X) - d(X,X) over all cuts (X,X) of D where d and c are lower and upper capacity functions. A natural

question to ask is what is the minimum flow in such a network. In order to answer this, consider a " dual " network D'(N,A) such that c' and d'

are capacities with -c(x,y) = c'(x,y)

d'(x,y) = -d(x,y). The maximum

flow in D' equals the minimum of d’(X,X)-c’(X,X). But the maximum flow in D' also equals the minimum flow in D while the minimum of

d'(X,X) - c1(X,X) is merely the maximum of d(X,X) - c(X,X). Thus

106

Corollary 6-9: If an integer flow function exists on a finite

network with d(a) < f ( a)

c(a) for all a e A where d and c are

integer valued functions then the minimum flow equals the maximum

d(X,X) - c(X,X) taken over all cuts (X,X) in D. Indeed

there is a minimum flow function which is integer valued.

Indeed here and in theorem 6-8 the existence of any flow function implies the existence of an integer valued flow function.

Theorems 6-6 and 6-7 are premised upon the existence of a flow in a network. When one is given lower and upper capacity functions in a

network, it is natural to seek conditions under which a flow exists. This question is resolved by the elegant result known as the circulation theorem due to Hoffman. The following discussion presents the circula­

tion theorem and explores its relationship to the integrity theorem. The circulation theorem deals with a network in which there is

no distinguished source or sink. On such a network lower and upper capacity functions d and c respectively are defined such that d(x,y) d(X,X).

107

Hence c(X,X)

d(X,x) for all X Ç N is a necessary condition for a

circulation to exist. The circulation theorem states that this cond^ ition is sufficient.

Before proceeding to the circulation theorem itself, it will he

useful to consider how one may pass from a network on which a circul­ ation is defined to one in which a nonnegative flow exists. Namely,

given a source and sink free network D(N,A) with a circulation f ,one

can augment this network with a source and a sink in the following manner. Let N' = N U{s,t} and A' = A j {(s,x),(x,t) ] x E N} .

Define a capacity function c1 on A1 as follows

c’(x ,y) = c(x,y) - d(x,y) for all (x,y) E A c1(s,x) = d(N,x)

for all x £ N

c *(x,t) = d(x,N)

for all x E N.

Then the circulation f in the original network corresponds to a flow f1 in the augmented network as given by f’(x,y) = f(x,y) - d(x,y) for all (x,y)

f’(s,x) - d(N,x)

for all x E N

f'(x,t) = d(x,N)

for all x E N.

e

A

For example, figure 6-9 illustrates a network with a circulation of value 2 on all arcs while figure 6-10 illustrates the augmented network with source s and sink t. ( Again the previously prescribed convention of indicating the ordered triple d,f,c with d < f < c

next to each arc is adhered to. )

108

Figure 6-9

2

z

Figure 6-10

There is a slight problem of non-negativity in the above construc­ tion when some of the lower capacities are negative. This, however, is

easily rectified by taking c’(s,x) = f’(s,x) = d(N,x) + k and c'(x,t) = f’(x,t) = d(x,N) + k where k is a sufficiently large positive constant.

Having established this connection between circulations and flows in networks one can now apply the max-flow min-cut theorem ( or the integrity theorem ) to yield the circulation theorem.

Theorem 6-10: ( A.J.Hoffman ) Let D(N,A) be a source-sink free

network with lower and upper capacity functions d(a) and c(a), respectively. Then there exists a circulation in D(N,A) if and

only if

c(X,X) > d(X,X)

for every X C N.

109

Proof:( Ford and Fulkerson ) The necessity has already been

demonstrated. Now suppose c(X,X) > d(X,X) for every X

N. Then a

circulation in D( N ,A ) will exist if and only if there is a flow of

value f'(s,N) = d(N,N) in D(N’,A*). But for a flow of value d(N,N)

to exist the max-flow min-cut theorem implies that it is

necessary

and sufficient that all cut capacities exceed d(N,N); ie , c'(X',X')

d(N,N) for all cuts (X’,X’) in D(N',A'). But this last

inequality does hold. For if (X’,X’) is any cut with X=X'-s and

X=X'-t then cT(X*,X') = c*(X u s, X u t) = c'(X,X) + c'(s,X) + c'(X,t)

= c(X,X) - d(X,X) + d(N,X) + d(X,N) = c(X,X) - d(X,X) + d(X,X) + d(X,X) + d(X,N)

= c(X,X) + d(X,X) + d(X,N) = c(X,X) - d(X,X) + d(N,N) > d(N,N). ( This shows that a flow of value at least d(N,N) exists. But the capacities c'(s,x) show that the maximum flow is at most d(N,N) and

that in a maximum flow f'(s,x) = d(N,x) and f'(x,t) = d(x,N) for all

x e N. From this, f(a) = f1(a) - d(a) is easily verified to be a

circulation ). I

Theorem 6-10 was proved for real valued capacity functions and employed the max-flow min-cut theorem. One can see from the proof that

given integer valued capacity functions, then the integrity theorem implies

the discrete version of the circulation theorem ( which asserts the

existence of an integer valued circulation ).

110

One can also pass from a given network in which there is a flow to an equivalent network in which there is a circulation. Namely,

adjoin an arc from the sink t to the source s with c(t ,s) = +»

where +°°

represents an integer of large magnitude which imposes

no constraint on the flow in arc (t,s ) of the network. Mow the value of the flow in the original network becomes the value of f(t,s) in

circulation of the augmented network. ( There is a possible incon­ venience of an are (t,s) in the original network but this will be

ignored since there are simple ways of surmounting this difficulty. ) Now by considering this construction it becomes clear that the

circulation theorem ( integer case ) yields the integrity theorem in the following manner. To the augmented network of D(N,A) described

above let d(t,s) = c

where c is the minimum cut capacity in D. Then

any circulation in the augmented network corresponds to a flow in D(N,A)

with v greater than or equal to the minimum cut capacity c. This clearly yields the nontrivial part of the integrity theorem. To justify the use of the circulation theorem, ie , to produce a

circulation , it must be established that c(X,X) - d(X,X) _> 0 for all

X

N. To this end let d(x,y ) = 0 for all (x,y ) E A. Then there are

four cases to consider :

( i ) if s E X and t E X, then c(X,X) _> 0 and d(X,X) = 0 and the inequality holds ;

(ii) if s E X and t £ X, then c(X,X)_> 0 and d(X,X) = c, hence

c(X,X) - d(X,X) = c(X,X) - ?

> 0;

Ill

( ill ) if s

X and t E X then c(X,X) = 00 and since c(t,s)

, the

inequality holds ; (iv) if s I X and t £ X then c(X,X) _> 0 and d(X,X) = 0 and again

the inequality holds.

Thus the integrity theorem and the circulation theorem are completely equivalent in their discrete or continuous forms.

Theorem 6-11 : The integrity theorem is eauivalent to the circulation

theorem.

The discussion of the circulation theorem began with the question

of determining conditions under which a feasible flow exists in a given network with an upper and lower capacity function. If one adjoins an arc (t,s) to the network with d(t,s) = -°°

( ie, no constraints ) then

the existence of a flow in the original network is equivalent to the existence of a circulation in the augmented network. This would require c(X,X) - d(X,X) > 0 in the augmented network. Now the last inequality

is automatically satisfied if s E X and t E X or if s £ X and t E X. Thus the inequality need only hold in the remaining two cases. This

yields

Theorem 6-12: Let D(N,A) be a network with source s and sink t

and with upper capacity c(a) and lower capacity d(a). Then there

is a flow function on D(N,A) with d(a) f(x,N) and c(x) > f(N,x) for x e N. Notice if x f s,t then

f(x,N) = f(N,x) and so both inequalities are saying the same thing. Now a cut set of nodes is a set S c N whose removal from D(N,A) would leave no s,t are path. ( Of course, any S containing s or t is a cut

set ). The capacity of a set of nodes is defined as c(S) = £ c(x). X£S The node form of the integrity theorem for non-negative flows becomes :

Theorem 6-13: ( Integrity theorem - node form ) The maximum flow

\) through a network equals the minimum capacity of a node cut of D(N,A). In addition, if the capacity function is integer then there is a maximum flow

function which is integer valued.

Proof: ( Ford and Fulkerson ) Modify any network D(M,A) by replacing

113

V E N with a pair of vertices v. and vr ( called the left image and the right image of v ) and an arc (vg,v ) joining them. Also, the arc

a = (vi(vp denoted by

e

A is replaced by an arc Joining the right image of

,

, to the left image of Vp denoted by v^ . Assign an

infinite capacity to each are replacing a E A and a capacity c(v) to

(v^,vrL The are form of the integrity theorem yields the desired

result.I

i.gure o-ll illustrates the procedure described above.

Figure 6-11

Starting with the node form of the integrity theorem the arc form, theorem 6-5, is just as easily established.

Theorem 6-14 : The node form of the integrity theorem implies

the arc form.

Proof : Modify D(W,A) by introducing dummy nodes v& on each a e A. By assigning infinite capacities to each node v £ N and capacities c(a) to va, the ca.pacity function can be viewed as a nonnegative integer function on the arcs of D(N,A). Applying the node form of the integrity theorem yields the arc form.I

114

The node form of the integrity theorem implies a mixed form where capacities are assigned to both arcs and nodes. Here, a cut is a set of nodes

and arcs whose removal from D(N,A) would leave no st arc path in the resulting

network. The capacity of such a cut is the sum of the capacities on the nodes and arcs comprising the cut. The mixed form is demonstrated by modifying D(N,A) as in theorem 6-14 but in this version capacities are assigned to both the nodes v e N

and the dummy nodes v&. Thus theorem 6-13 yields the mixed form

of the integrity theorem for non-negative flow functions.

Theorem 6-I5; ( Integrity theorem - mixed form ) The maximum flow v through a network D(N,A) equals the minimum capacity of a cut set of

nodes and arcs. In addition a non-negative integer capacity function

implies the existence of a maximum flow function which is integer valued.

Clearly the mixed form of the integrity theorem implies the arc form or the node form as can be seen by assigning infinite capacities to the nodes

or the arcs of D(N,A) respectively. Thus the above three forms of the integrity theorem are equivalent.

It should be pointed out that the node form and the mixed form of the

integrity theorem have generalizations analogous to those in theorem 6-8 and corollary 6-9 where both lower and upper capacity functions are defined and the non-negativity condition of the functions defined on both arcs and

nodes are relaxed. Likewise one can make a passage from the discrete versions

to the continuous counterparts as indicated in the remarks following theorem 6-5.

115

Although there is, as just mentioned, a node form analogous to theorem 6-8 and corollary 6-9 it turns out that the node form is not quite so straight forward and is rather awkward to express.( This might seem surprising in

view of the node form of the integrity theorem ( theorem 6-13 ); however, the ease in making the transition from the arc form to the node form was due

to the fact that the flow functions were constrained to be non-negative. The removal of this constraint causes various complications ).

Rather than provide a full generalization of theorem 6-8 to a node form,

it will suffice for present purposes to present the following limited form. This form entails a notion of comparability of the nodes of a network.

Namely, two nodes x and y of a network are called comparable if there is a

directed path from x to y or vice versa. Otherwise the nodes x and y are called incomparable. Then a limited node version of theorem 6-8 becomes :

Theorem 6-16: Suppose that an integer flow function exists constrained by an upper capacity function c defined on the node set N of a finite

network D(N,A). Then the maximum flow v through the network equals the minimum capacity of a node cut consisting of incomparable nodes.

The proof of theorem 6-16 utilizes a construction similar to that found in theorem 6-13» However, here in the modified network ( obtained by replacing

each node by a left and right image ) every lower capacity to be defined on arcs between left and right images of a node has a value of -00. This means

116

that in the modified network, there cannot be an arc of the form (X,X) ( otherwise theorem 6-8 would yield an infinite value ). The condition

that there can be no are from X to X

in the modified network translates

into the condition that no two nodes of the node cut be comparable. The proof of corollary 6-9 from theorem 6-8 was obtained by negating

all quantities involved. In a similar way the following corollary of

theorem 6-16 is obtained :

Corollary 6-17 : Let d be a lower capacity function defined on the node set N of a finite network D(N,A). If a flow function exists

such that d(x)

f(x) for all x e N then the minimum flow equals the

maximum of d(X) taken over all cuts of incomparable node subsets of N.

117

The fact that the integrity theorem is completely equivalent to Wenger's theorem is not surprising. Indeed, the integrity theorem can also be used rather directly to imply Konig's theorem ( hence Hall's

theorem ) and Dilworth's theorem. But a more unexpected result is that Hall's theorem

( at least in a weighted form ) can yield the integrity

theorem. The remainder of this chapter is devoted to showing the extent to which these theorems are related. First Wenger's theorem and the

integrity theorem are shown to be equivalent. Then the integrity theorem

( at least various forms and corollaries of it ) is shown to imply the theorems of Konig and Dilworth. Finally an extension of Hall's theorem

( theorem 3-15 ) is used to imply the circulation theorem and hence

the integrity theorem. This elegant result is due to A.J.Hoffman.

Theorem 6-28 : The integrity theorem implies Wenger's theorem.

Proof : Let G ( V, E ) be any finite graph with vertex set V and edge set E. Let u and w be nonadjacent verticies of V(G). Wodify G(V,E) so that every edge incident with u is assigned an outward direction ( points away from u ) and every edge incident with w is assigned an

inward direction ( points towards w ). In addition,.replace all other edges of E by a pair of oppositely oriented arcs. Assign an infinite

capacity to u and w and a unit capacity to all other nodes of V. The

node form of the integrity theorem implies the maximum flow x> from

u to w equals the minimum capacity of a node cut separating u from w.

118

Since all nodes of V other than u and w have unit capacities, the maximum integer flow f is a 0,1 function which yield V independent uw paths. Thus the minimum cardinality of a uw separating set equals

the maximum number of uw vertex independent paths.I

The converse of theorem 6-18 is obtained by applying the directed multigraph-edge version of Monger's theorem ( theorem 5 - 1Ô •

Theorem 6-19: Monger's theorem implies the integrity theorem.

Proof: Let D(N,A) be any directed network with source s and sink t. Let ca be an integer capacity assigned to each arc of A. Modify the network by replacing each are a of A by cQ arcs oriented similarly

to a. Thus the network is replaced by a directed multigraph. By the edge form of Monger's theorem, the minimum cardinality of an edge

separating set equals the maximum number of edge independent paths from

s to t. Clearly, if one edge of a multiedge is in the minimum st edge separating set, every edge must be. Now the minimum number of edges in an

s -t edge separating set corresponds to a minimum capacity of a cut and the maximum number of st independent edge paths corresponds to a

maximum flow

from s to t in D(N,A). Hence the minimum capacity of

a cut equals the maximum flow from s to t.I

119

Theorem 6-20 : The integrity theorem implies Konig’s theorem.

Proof: Let G(X,E,Y) be any finite bipartite graph. Modify G by introducing a source s with arcs (s) for each x^e X and a sink t

with arcs (y^ t) for each y< e Y. Impose a direction from X to Y on

all edges of E. Let c be a capacity function assigning an infinite capacity to s and t and a unit capacity to every element of X and Y.

This procedure is illustrated in figure 6-12.

Modified to

s 1

Figure 6-12

By the node form of the integrity theorem, the minimum cut capacity v equals the maximum flow from s to t. The minimum vertex cover of

G(X,E,Y) corresponds to a minimum cut of the network. In addition, the maximum ( integer-valued ) flow must be such that f(e) = 1 on

v independent edges of E. For otherwise it contradicts the fact that each node of X and Y has unit capacity. Thus the minimum sized vertex cover of G equals the size of a set of independent edges of G

and consequently equals a maximum sized independent edge set. I

120

Theorem 6-21: The integrity theorem ( in the form of corollary 6-17 ) implies Dilworth’s theorem.

Proof: Let < P, £ > be a finite partially ordered set. From < P, _< > , form a network D(P,A) by letting (x,y) e A if and only if x < y. Let all

minimal elements of < P,
be sources and all maximal elements of

P, < > be sinks in D(P,A). ( Alternatively one could adjoin to the

network a source s and arcs from s to all maximal elements along with a sink t and arcs from all minimal elements to t ). Assign a lower

capacity of 1 on all nodes of D(P,A). Then the minimal integral flow v

through the D(N,A) equals the maximal cardinality of a node cut in

D(N,A) consisting of incomparable nodes. But the minimum flow must cover each node in D(N,A) and corresponds to

v chains covering < P, £ > .

In addition, the maximum cardinality of a cut in D(N,A) consisting of incomarable elements corresponds to the size of a maximum antichain

in < P, _< > . Hence the minimum number of chains covering < P, £ > equals

the maximum size of an antichain in < P, < >

I

Next the relationship between Hall’s theorem and the Circulation

theorem is explored. For this purpose consider any source-sink free network D(N,A) with lower and upper capacity functions d and c respect­

ively. Then the nontrivial part of the circulation theorem states that

d(S,S) jc c(S,S) for all S C N implies there exists a circulation function

121

f defined on the arc set A of the network D(N,A); that is, there

is a circulation function f such that d(a) a C A. Now a

f(a)

c(a) for all

suprising fact is that the edge weighted version

of Hall’s theorem, theorem 3-15, implies that this condition

d(S,S) 5. c(S,S) for all S C N is sufficient for a circulation to exist in D(N,A).

Theorem 6-22: ( Hoffman ) The edge weighted version of

Hall's theorem implies the Circulation theorem.

Proof: Let the node set of D(N,A) be N = { x , ... , xn }

and let c(i,j) and d(i,j) be upper and lower capacities on the arc ( x^ ,Xj

. Without loss of generality one can assume that A

consists of all ( X|,xj ) with i

j by letting c(i,j) = d(i , J ) = 0

on all superflous arcs,

Now construct the complete bipartite graph G(N",E,N^) whose left and right sides are disjoint copies of N. Then if xv E N- is the left copy of x^ e N and Xj e N* is the right copy of Xj £ N,

then the edge x^Xj is denoted by e^j. Let M be a large integer in comparison to c(i,j) and d(i,j)

( i.e., let M be larger than the sum of all of the absolute values of the c(i,j) and d(i,j) ). Then assign vertex weights to G, namely a^

=

M - d(i,N) is assigned to x^

bj

=

M - d(N,j) is assigned to xj

( Then it is certainly true that

Z ai X- e N-

=

Z b« ). xj e «

122

Now in order to apply theorem 3-15, one assigns edge constraints

to the edges of G. Namely, let h(i,j) = c(i,j) - d(i,j) for i f J and h(i,i) = “ (Let « stand for an integer much larger than all the a^ and b j, say, larger than the sum of them).

Now assuming the sufficiency condition of the Circulation theorem, namely d(S,S)
L(G) M(V,E) D(N,A) M(N,A) V _ (X,X) c(X,X),f(X,X),d(X,X) A A? xty AIJ Xj Bl

r

page 9 9 9 9 9 10 10 10 10 13 13 15 15 15 16 16 18 18 16 20 20 20 21 21 UT hl hl 52 56 6h 8h 86 89 91 95 97 97 128 128 128 131 131 131

154

List of Name s G. R. R. G. E. P. J. A. L. G. D. D. T. P. M. R. F. I. A.

Birkhoff A. Brualdi P. Dilworth A. Dirac Egervary Elias Farkas Feinstein R. Ford Jr. Frobenius R. Fulkerson Gale Gallai Hall Hall P. Halmos Harary Heller J. Hoffman

D. H. L. K. L. H. M. J. R. J. G. C. W. A. H. H.

Konig Kuhn Lovasz Menger Mirsky Perfect A. Perles Petersen Rado T. Robacker E. Shannon B. Tompkins T. Tutte W. Tucker Tverberg E. Vaughn

page 36, 38, 39, 137 22 4, 64 5, 8, 26, 29, 76, 77, 80, 90 3, 37 84 131 . 84 4, 6, 8, 70, 71, 91, 93, 99, 100, 112 3, 35 4, 6, 8, 70, 71, 84, 91, 93, 99, 100, 112 130, 135 22 1, 40 131 4, 8, 40, 42 82 7, 135 6, 55, 106, 108, 112, 117, 121, 124, 134, 135, 136, 137, 138 2, 8, 25, 39, 78, 79, 80, 137 6, 7, 130 4, 8, 29, 30 3, 76, 78 40, 70 4o, 70 4, 8, 64, 66, 68, 8o 7, 26 4, 8, 40, 43 79 84 7, 135 7 130 4, 8, 64, 66, 68, 80 4, 8, 4o, 42

155

Index

acyclic graph adjacent alternating path antisymmetric antichain antichain length antichain cover arc arc separating set are separating set, strong arc separating set, weak are set augmented network augmented path

15 10 26, 27, 28 6h 66 66 66 89 91 91 91 93 107, 110 99, 10U

basic feasible solution bipartite graph bivalent broken path

127 2, 18, 25 17 72

capacity function chain chain cover chain length circulation circulation theorem circulation theorem, proof common system of representatives, GSR comparable comparable nodes _ complement of a graph, G complete bipartite graph, Kn m complete graph, Kp ’ complete matching connected connected component cover, antichain cover, chain cover, edge cover, vertex convex linear combination convex polygonal region coset of a subgroup

96, 97 4, 65 66 66 106 6, 8, 56, 108, 120 108 52 4, 64 115 16 18 16 20 15 15 66 66 21 20 38 126 2, 55

156

Cramer’s rule cuts cut capacity cycle, Cn

135 97 97 15

degree of a vertex, deg(v) deficiency 6 dependent sets Dilworth’s theorem Dilworth’s theorem, proof directed graph, D(N,A) directed multigraph, M(N,A) directed network disconnected doubly stochastic matrix dual of Dilworth’s theorem dual of Dilworth's theorem, proof dual family dual of Konig’s theorem dual of Konig's theorem, proof dual of Menger's theorem dual problem duality theorem of linear programming

10 45 45 4, 5, 6, 8, 64, 66, 67, 69, 71 73, 76, 93, 105, 117, 120 67 86, 88, 89, 93, 97 86, 88, 91, 118 93 15 36,37 64, 74, 75 75 47 25,32,33 33 84 129 6, 124, 130, 131, 140

edge edge edge edge edge

9 15 15 84 58

covering path separating set walk

feasible solution finite character matroid first subdivision of a graph flow function flow

126, 129 44 87 95, 97 95

Gallai's theorem Gallai's theorem, proof graph, directed D(N,A) graph, multi M(V,E) graph, multidirected M(N,A) graph, simple graph, trivial

23, 33 23 89, 93 86 91 9 16

157

half planes Hall's condition Hall’s condition, weighted Hall's theorem

Hall’s theorem, proof handshake lemma Hoffman's condition

,

incidence matrix incident incomparable incamparable nodes independent ars paths independent edge paths independent edges independent edge set independent entries of matricies independent node paths independent sets independent vertices independent vertex paths independent vertex set induced subgraph Integrity theorem Integrity theorem, Integrity theorem, Integrity theorem, Integrity theorem, isolated vertex isomorphism invariants

are form mixed form node form proof

126 1, 41, 48, 70 59, 93, 121 1, 4, 5, 6, 8, 40, 42, 64, 76 93, 117, 120, 138 42 11, 17, 94 135, 136 33, 37, 50, 73 10 64 115 91 84, 85 19 20 34 89 45 19 76 20 13 5, 6, 8, 84, 86, 91, 93, 99, 100 106, 112, 117, 119, 120 112 114 112 100 16 12, 37 12

join

10

k-valent Konig's theorem

Konig's theorem, proof Koni g-Egervary theorem Konig-Egervary theorem, proof

16, 48 2, 4, 5, 8, 27, 33, 40, 64, 69, 71 73, 76, 80, 82, 93, 117, 119 27 3, 8, 34, 35, 38, 73, 125, 138, 140 34

Length of an antichain length of a chain length of a walk line

66 66 14 34

158

line covering line graph, L(G) linear constraints linear programming linear programming problem loops lower capacity function,d

3b 8b 126 6, 12b 125, 126 87 103

marriage theorem matching matching X-complete ( Y-complete) matching, perfect matching theorems Matching theorem Matching theorem, proof Matching theorem, Weighted matroids matroid theory Max-Flow Min-Cut theorem Max-Flow Min-Cut theorem, proof maximal antichain maximal chain maximal element of < P, > maximal independent edge set maximal independent vertex set maximum degree of a graph,A maximum element of < P, > maximum edge separating set maximum flow maximum flow function, v maximum of the sizes of the independent edge sets,6, maximum of the sizes of the independent vertex sets ,B0 Mengerian theorems Menger's theorem

b8 2, 19 b8 bT 1 2, 25, b8 b8 56 bb 7 4, 5, 6, 93, 99, 109 99 66 66 65 20 20 10 65 85 98, 102 97

Menger•s theorem, proof minimal edge covering set minimal element of < P, minimal vertex cover minimum cut capacity minimum degree of a graph,6 minimum edge separating set minimum element of < P, minimum flow

20

20 1 3, b, 5, 6, 8, 25, 26, 76, 77 80, 82, 86, 88, 90, 93, 117 77 21 65 21 98, 102 10 85 65 105

159

minimum of the sizes of the edge covering sets, ai 21,28 minimum of the sizes of the vertex covering sets, ao multigraphs, M(V,E) multidirected graphs, M(N,A) multiple edges multiple sinks multiple sources

21,28 86 86 86 103 103

n-cycle network nodes node path node set node separating set nonadjacent vertexsets nonbasic optimal solutions nonnegativity condition

15 93 89 89 93 89 83 128 126

one-sided vertex cover objective function optimal feasible solution output

21, 29 126 126, 129 9U

partially ordered set, poset partial transversal path predecessor perfect matching permutation matrix primal problem

4, 6^ 45 15 64 20 36, 137 129

rank function, p reachable reachable arc reflexive regular graph

45 28 99, 103 64 16

saturated saturated arc second subdivision separating set of arcs separating set of edges simple finite graph

19, 26 96 60 97 85 9, 90

160

simplex method sink source spanning subgraph stochastic matrix strong are separating set strong edge separating set strong vertex separating set strongly independent arc paths strongly independent edge paths strongly independent vertex paths subgraph system of representatives, SR system of distinct representatives, SDR s-z augmented path

6, 13b 93, 97 93, 97 13 37 91, 92 86 83 91, 92 86, 88 79 12 bl bl, 70 99

totally disconnected graph transitive transversal transversal theory trivalent trival graph type (i),(ii) arcs type 1 (2) paths

16 6b bl 2, bO 16 16 99, 102 81

unimodular unimodularity condition unreachable unsaturated upper capacity function, c

13b, 135, 136 12b 28 26 103

value of the flow, V vertex vertex covering vertex path vertex separating set vertex walk

95 9 9 lb 76 15

weak are separating set weak edge separating set weak node separating set weak vertex separating set weakly independent arc paths weakly independent edge paths weakly independent node paths weakly independent vertex paths Weighted Hall condition

91, 92 86, 88 89 79 91, 92 86 89 83 59

161

Weighted Matching theorem, proof

59

zero flow

96