MENGER'S THEOREM AND MATROIDS

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1 MENGER'S THEOREM AND MATROIDS R. A. BRUALDI 1. Introduction Let G be a finite directed graph with X, Y disjoint subsets of the nodes of G. Menger's theorem [6] asserts that the maximum cardinal number of a set 0 of pairwise node disjoint paths from X to Y equals the minimum cardinal number of a set which separates X from Y. A special case of Menger's theorem occurs when all the edges of G have initial node in X and terminal node in Y (the bipartite situation). The resulting special case is the well-known Konig duality theorem [5]. In [1,2 and 3] generalizations of Konig's theorem were obtained by assuming that X and Y had matroids defined on them and then requiring that the set of initial (resp. terminal) nodes of 0 be independent in the matroid on X (resp. Y). The case where Y only had a matroid defined on it had already been handled by Rado [8]. Our purpose in this note is to raise Menger's theorem to the same level of generality. 2. Matroids We present here those aspects of matroid theory which are required for the statement and proof of our main result. Let be a finite set. A matroid [11] 8 on E is a non-empty collection of subsets of E, called independent sets, satisfying: (i) Ae$, A' L A imply A'e&, (ii)a l,a 2 e& i I-/4J + 1 = /I 2 implythereisan A;6y4 2 \-4 1 with^ u {x}es. Subsets of E which are not in & are called dependent sets. Each subset A of E has a welldefined rank, r(a), which is equal to the common cardinal number of all maximal independent subsets of A. The rank of the matroid $ is r{e). A maximal independent subset of is a basis of S. For A c E the closure of A is defined by: Put another way, the closure of A consists of the elements of A along with those elements of E\A which depend on A. If cl (A) = A, we call A a.flat (or a closed set). Thus if A is closed and y A, r(a u {y}) = r(a) +1. If $ is a matroid on E and F ^ E, then $ F = {A F : A e S) is a matroid on F, called the restriction of $ to F. Obviously the rank functions of and 8 F agree on subsets of F; we will not distinguish them. Let B be a basis of $E\F- Then $ F = {A ^ F \ AKJ BsS) is a matroid on F, called the contraction of $ to F. The contraction & 0F does not depend on the choice of basis B of & \ F (see [1] and also [2]). If r F denotes the rank function of F, then = r{akj{e\f})-r{e\f), for all A^F. If E u E 2 are disjoint sets and S u t are matroids on E t, E 2 respectively, then S y & 2 - {^i v A 2 : AiG&i (i = 1, 2)} is a matroid on JE ± u E 2, called the direct Received 18 June, Research supported by a N.A.T.O. postdoctoral fellowship at the University of Sheffield. [J. LONDON MATH. SOC. (2), 4 (1971), 46-50]

2 MENGER'S THEOREM AND MATROIDS 47 sum of i x and S 2. If F t E t (i = 1, 2), then r(f x u F 2 ) = r^f^+r 2^) where r, r 1, r 2 denote the rank functions of ^ <? 2, S u S 2 respectively. If is an arbitrary finite set, then the collection &{E) of all subsets of is obviously a matroid. 3. Main result Let G be a finite directed graph whose set of nodes is N. By a path 9 in G we mean a linearly ordered sequence (x u x 2,..., * ) of n ^ 2 distinct nodes of G with (x f, A',+I) an edge of G (1 ^ i < n). The initial node of 0 is x l5 denoted by In 0, while the terminal node of 0 is x n, denoted by Ter0. If 0 is a collection of paths, then In 0 = {In0 : 0 e 0} and Ter 0 = {Ter0 : 9 e 0}. The set of nodes of the path 9 = (xj, x 2,..., x n ) is Nod0 = {x lt x 2,..., x n }. If 0 is a set of paths in G, then 0 is pairwise node disjoint provided the sets Nod 9 (9 e 0) form a pairwise disjoint collection of sets. If 9 l = (x l}..., x n ) and 9 2 = (x n,..., x m ) are paths having only the node x n in common, then 9 X.9 2 is tne P atn ( x u > ^n» > **.) Let X and 7 be sets of nodes of G and let <^>1, ^2 be matroids on X, Y with rank functions r 1, r 2, respectively. In what is to follow we are to be interested in paths from X to Y (paths with initial node in X and terminal node in Y). There is no loss in generality if we then assume that X and Y are disjoint sets with each edge (z ls z 2 ) with {z u z 2 } r\x T 0 satisfying z x ex,z 2 $X and each edge (z ls z 2 ) with {z t,z 2 }n Y # 0 satisfying z x $ Y, z 2 e Y. This is because we may select sets X* = {x* : x e X}, Y* = {y*:yey} where \X\ = Z*, Y = \Y% and X*, 7*, N are pairwise disjoint, and consider instead of G the graph G* whose set of nodes is N u AT* u 7* and whose set of edges consists of those edges of G along with the edges {{x*,x):xex}u{(y,y*):yey}. We may then carry the matroid structures on X, 7 over to X*, 7*, respectively. We thus make the above assumption on G so that, in particular, every path from X to 7 has only its initial node in X and its terminal node in 7. A subset Z of the nodes of G is called a separating set (for the sets X, 7 with matroids S x, S 2 respectively) provided every path 9 from X to 7 satisfies {Nod 9} r\z # 0. The index of the separating set Z is H(Z) = r\z n X)+r\Z n Y) + \Z\{X u 7}. The definition of a separating set could be modified by insisting in the above definition that Z n X is a flat of the matroid i x on X and that Z n 7 is a flat of <T 2 on 7. It could also be modified by saying that Z is a separating set provided In0 ecl^z n X), Ter 0 e cl 2 (Z n 7) or {Nod0} n Z # 0 for all paths 0 of G from Z to 7 (cl' denotes the closure operator for <T (i = 1, 2)). No matter what definition is used the following theorem is true. THEOREM. The maximum cardinal number of a set 0 of pairwise node disjoint paths from X to 7 with InOe^1 and Ter0e<f 2 equals the minimum index of a separating set. In the proof presented below we shall be guided by both Pym's recent simplification [7] of Dime's proof [4] of Menger's theorem and the proof of the result for the bipartite situation as given in [1] and [3].

3 48 R. A. BRUALDI Let 0 be a collection of pairwise node disjoint paths from X to Y with In Ter0eS 2, and let Z be a separating set with Z 1 =ZnX, Z 2 =Zr>Y, Z o = Z\{Z t vz 2 ). Let 0! s 0 consist of those paths 0 with In0eZ 1} 0 2 g0 consist of those paths 0 with Ter0eZ 2, and 0 o = 0\{0 x u 0 2 }. Since In < * I0J < T X {Z^), and since Ter Ge^2, 0 2 ^ r 2 (Z 2 ). All paths in 0 o must contain a node of Z o ; since paths in 0 and hence in 0 O are pairwise node disjoint, 0 O < Z 0. Hence l l < l il + l 2l + l ol < r l (Z 1 )+r 2 {Z 2 ) + \Z 0 \ = /i(z). Thus if A: is the minimum index of a separating set, to complete the proof we need only show that there is a set of k pairwise node disjoint paths from X to Y whose initial (resp. terminal) nodes comprise an independent set of 8 x (resp. S 2 ). We prove this by induction on the number of edges of G. If G has only one edge, this is readily verified. Otherwise we distinguish two cases. We may assume each node of X and each node of Y is incident with at least one edge of G. Case 1. Every separating set Z with index k satisfies Z = X or Z = Y. Let x, z be nodes with xex and (x,z) an edge of G. Consider the graph G' obtained from G by removing the edge (x, z) only, and consider still the sets X, Y and matroids S 1 and S 1. If all separating sets W relative to G' have index p'(w) ^ k, then the conclusion follows by induction. Otherwise there is a separating set W relative to G' with n'{w) < k. But then W VJ {x} is a separating set relative to G with H(W u {x}) < k; we must have equality and since xex, W u {x} <= X. Likewise W \J {y} is a separating set relative to G with index k, and since z$x, W KJ {Z} S Y. Since Xn7 = 0, this means W = 0, k = 1, {x}e l, {z}e 2. The path (x,z) is of the type desired. Case 2. There is a separating set Z with index k with Z ^ X, Z # Y. Let Z l =ZriX,Z 2 =Zr\Y, and Z o = Z\{Z X uz 2 }. Let G 1 be the graph consisting of those nodes and edges on paths 6 of G with InfleX! = X\Z lt TerQeZ 0 vz 2, and no other nodes in common with X\Z t or Z o uz 2. The graph G 1 has fewer edges than G since no edge with terminal node in Y\Z can belong to G 1. Consider the disjoint sets X t and Z o u Z 2 with matroid <f Xl on X t and matroid ^(Z o ) < f 2 on Z o u Z 2. Suppose, relative to these considerations, there were a separating set A with index n l (A) < k r^iz^). Consider A\JZ± and let 9 be any path in G from X to 7. If Intf^ and Ter0^Z 2, then {Nod 0} nz o?t 0; hence an initial segment of 0 is a path from X t to Z o in G 1 so that {Nod0} n{ This means ^4 u Z t is a set separating X from 7; moreover = r 1^ n X} uzj+r 2^ n Y) + 4\{* u y} = r 1 Xl(AnX)+r\Z 1 )+r\an Y) + \A\{Xu Y}\ This is a contradiction. Hence by induction there is a set Q t of paths in G 1 with In ^^^, and Ter 0 X e ^(Z o ) «^ 2. In particular Ter i = Z o u B 2,

4 MENGER'S THEOREM AND MATROIDS 49 where B 2 is a basis of <^ 2. It follows by an analogous argument that if G 2 is the graph consisting of those nodes and edges on paths 0 in G with InfleZj uz 0) Ter 0 e Y 2 = ^\Z 2 and no other nodes in common with Z t u Z o or Y\Z 2, then there is a set 0 2 of k r z (Z 2 ) = r x {Z 1 )-\-\Z 0 \ pairwise node disjoint paths in G 2 with In 0 G 2 &zi ^( z o) and with Ter 0 2 e % Y2. In particular In 0 2 = J3 X u Z o where B x is a basis of S^. The paths in Q 1 and 0 2 can have only the nodes of Z o in common; otherwise the separating property of Z is violated. The collection 0 = {9 1 & i 1 2 } { } {9,.0 2 : 0! G i, 0 2 e 0 2, Ter0 x = In0 2 ez o } is a collection of pairwise node disjoint paths in G with In 0 = B x u In 0 t e g 1 and Ter 0 = B 2 u Ter 0 2 G 8 2. This completes the proof. If it is assumed that X n 7 = 0 and that the only nodes of paths in X and Y are the initial node in X and terminal node in Y, then the notion of a separating set needs to be replaced by that of a separating triple (Z 1,Z 0,Z 2 ) where Z x X, Z 2 ^ Y and every path 0 from X to Y satisfies In 0 G Z 1} Ter 0 GZ 2, or If the index of a separating triple is defined by ^1(Z 1 )+r 2 (Z 2 ) + Z 0, the preceding theorem remains valid provided degenerate paths consisting of one node are permitted. This extension can be derived from the preceding theorem by introducing new nodes and edges as explained previously. As an example consider the graph whose set of nodes is {x u x 2, y} and set of edges is (x u x 2 ) and (x 2, y) with matroid S 1 = {0, {xj} on X = {x u x 2 } and matroid S 2 = {0, {y}} on Y = {y}. If one uses the notion of a separating set as before, then {x 2 } is a separating set with index 0; on the other hand the path 0 = (x lt x 2, y) satisfies {In9}eS x, {Ter0} G S 2. Examples of separating triples with index 1 are ({xj, 0, 0) or (0, {x 2 }, 0) or ({x 2 }, {x 2 }, 0). We could have proved the theorem directly in this more general setting, but we have chosen not to. Another way to prove the theorem is as follows. If i z consists of those sets A X such that there is a set 0 of pairwise node disjoint paths from X to Y with In 0 = A, Ter 0 G S 2, then it is not difficult to show that $ 3 is a matroid on X. Thus in the theorem we are investigating the maximum cardinality of a subset of X which is independent in both S 1 and # 3. One can then use the method in [10] provided one can find a formula for the rank function of S 3. A special case of our theorem gives such a formula. It does not seem easier to prove this special case and pass to the general case as indicated above. (This method was suggested to me by D. J. A. Welsh.) From our theorem the rank function for S 2 (assuming X n Y = 0, etc., but with extensions as already indicated) is given by r\a) = min {r\z n Y) + Z\y }, where the minimum is taken over all sets Z separating A from Y. Finally we remark that the theorem remains true for infinite graphs. If there are no separating sets of finite index, then there are arbitrarily large collections 0 of paths of the type desired, while if there is a separating set of finite index, the theorem JOUR. 13

5 50 MENGER'S THEOREM AND MATROIDS remains true as given. The transition to infinite graphs can be accomplished in much the same way as Erdos has extended Menger's theorem to infinite graphs (see [5; pp ]). References 1. R. A. Brualdi, " Symmetrized form of R. Rado's theorem on independent representatives ", unpublished paper (1967). 2. f " Admissible mappings between dependence structures, Proc. London Math. Soc. (3), 21 (1970), , A general theorem concerning common transversals. Proceedings of the 1969 Oxford conference on combinatorial mathematics edited by D. J. A. Welsh (Academic Press) G. A. Dirac, " Short proof of Menger's theorem ", Mathematika, 13 (1966), D. Konig, Theorie der endlichen und unenlichen Graphen (Chelsea, New York, 1950). 6. K. Menger, " Zur allgemeinem Kurventheorie ", Fund. Math., 10 (1924), J. S. Pym, " A proof of Menger's theorem ", Monatch. Math., 73 (1969), R. Rado, "A theorem on independence relations", Quart. J. Math. Oxford Ser., 13 (1942), W. T. Tutte, " Lectures on matroids ", /. Res. Nat. Bur. Standards, 69B (1965), D. J. A. Welsh, " On matroid theorems of Edmonds and Rado ", /. London Math. Soc. (2), 2 (1970), H. Whitney, " On the abstract properties of linear dependence ", Amer. J. Math., 57 (1935), University of Sheffield, and University of Wisconsin.