Vertex-disjoint paths and edge-disjoint branchings in directed graphs

Vertex-Disjoint Paths and Edge-Disjoint Branchings in Directed Graphs

R.W. Whitty DEPARTMENT OF MATHEMATICAL SCIENCES

GOLDSMITHS’ COLLEGE, NEW CROSS LONDON, SE 14 6NW UNlTED KlNGDOM

ABSTRACT

A theorem of J. Edmonds states that a directed graph has k edge-disjoint branchings rooted at a vertex r if and only if every vertex has k edge-disjoint paths to r . We conjecture an extension of this theorem to vertex-disjoint paths and give a constructive proof of the conjecture in the case k = 2.

1. THE CONJECTURE

Let G = ( V , E ) be a finite directed graph with vertex set V and edge set E . Multiple edges are allowed, but self loops are excluded. An edge directed from x to y will be denoted by (x,y) (we do not distinguish between multiple edges from x to y), and we refer to x as the tail and y as the head of this edge. If P is a directed path from x to y , we refer to x and y as the tail and head of P , respectively, and say that P is trivial if x and y are the same vertex. If x’ and y ’ are the tail and head of a subpath of P , we write P :x’ + y ’ for the restriction of P to this path. Two paths are called edge-disjoint if they have no common edge and vertex-disjoint if they have no common vertices except possibly a common head or tail (the trivial path x is vertex-disjoint from precisely those paths with head or tail x). For a subset R of V , an R-branching in G is a spanning forest B of G in which all vertices of V - R have outdegree precisely 1. When R just consists of a single vertex r , we refer to B as an r-branching.

Let I = (1, . . . , k}, where 1 5 k 5 IVI. Let R = { R, 1 i E I } be a family of subsets of V, and let B = {B, I i E I } be a family of edge-disjoint branchings in G such that B, is an R,-branching. Let X = {x, I i E I } be a family of (possibly

Journal of Graph Theory, Vol. 11, No. 3, 349-358 (1987) 0 1987 by John Wiley & Sons, Inc. CCC 0364-9024/87/030349-10$04.00

350 JOURNAL OF GRAPH THEORY

nondistinct) vertices of V . Then X is called edge (respectively, vertex) -linked onto R if, for some permutation a : f -+ I , there exists a set of edge (vertex) -disjoint paths {P, i E I} such that P, has tail xi and head in R,a. If a and {P, ) i E I } can further be chosen so that P, lies in B,,, for i E I , then X is called B-edge (vertex) -linked onto R. In the case where R , = . . . = R, = { r } , we write (B)-edge (vertex) -linked onto r , for short.

1.1 Conjecture. The following are equivalent:

(1) Every family X = {xi I i E f } of vertices of V is edge (vertex) -linked

(2) There exists B such that every family X = {xi I i E I } of vertices of V is onto R .

B-edge (vertex) -linked onto R.

1.2 Example. Consider the graph G shown in Figure l(a). Let R, = { r , } , R , = {r2 , r,}, and R, = {r,, r4}. It is straightforward to ascertain that, for either edge- or vertex-linkings, G satisfies Conjecture 1.1( l), and that the R,-branchings shown in Figure I(b) satisfy Conjecture 1.1(2).

Conjecture 1 . 1 restricted to edge-linkings is a well-known theorem of Edmonds [ 11. A deep generalization has been given by Schriver [7] from which many other “edge-disjointness” results may also be derived, but which seems not to extend directly to questions involving “vertex-disjointness.” An efficient algorithmic proof of Edmonds’ Theorem has been given by Tong and Lawler [8]. In fact, for edge-linkings, Conjecture 1.1(2) is trivially equivalent to the existence of any family of k arbitrary edge-disjoint R,-branchings in G . For vertex-linkings this is false, as may be seen in Figure 1 by exchanging the edges (x , r , ) and (x, r3) between the first and second branchings, and by ascer- taining that the family {x,x,x} is no longer B-vertex-linked onto R .

Frank seems to have been the first to raise the possibility of an extension of Edmonds’ Theorem to vertex-disjoint paths (see [6] ) . He proposed the special case of Conjecture 1.1 where only r-branchings are considered and the families of vertices X are restricted to have only one distinct vertex (this is weaker than Conjecture 1.1 in the sense that there are graphs in which can be found a family B of r-branchings satisfying Frank’s conjecture but failing to satisfy Conjecture

@ ’4 3

DISJOINT PATHS AND BRANCHINGS IN DIGRAPHS 351

1.1). In this paper we prove Conjecture 1.1 for the case k = 2. The proof is rather ad hoc and does not seem to suggest a proof of the general case. On the positive side, it is constructive and may easily be seen to give a polynomial algorithm for constructing the required branchings.

We conclude this introduction by observing that Conjecture 1.1 is closely connected to problems concerning the orientation of undirected graphs to obtain various directed connectivity properties. Again, the known results focus on edge-connectivity, the strongest being those of Frank [3] and Nash- Williams [ 5 ] .

2. SIMPLIFYING THE CONJECTURE

In this section we show that, to prove Conjecture 1.1, we need only consider r-branchings and graphs in which every vertex except r has outdegree precisely k . These simplifications are essential for the techniques used in Sections 3 and 4 to prove the case k = 2. We first reduce Conjecture 1.1 to

1.1’ Conjecture. are equivalent:

Let G be a graph and r a vertex of G. Then the following

(1) Every family X = {xi 1 i E I } of vertices of G is edge (vertex) -linked

(2) There exists a family B of r-branchings in G such that every family onto r.

X = {xi I i E I } of vertices of G is B-edge (vertex) -linked onto r.

2.1 Lemma. Conjecture 1.1’ implies Conjecture 1.1.

Proof. Given a graph G = ( V , E ) and an arbitrary family R of subsets of V, we must construct a graph G ’ = ( V ’ , E ’ ) having a vertex r, such that Con- jecture l . l’( l) and (2) each hold in G’ for r if and only if Conjecture 1.1(1) and (2) hold in G for R. We construct G ’ in stages as follows: suppose G has m vertices, xo, . . . , x ~ - ~ , such that xJ lies in tf 2 1 of the members of R , 0 5 j 5 m - 1. We shall construct a series of graphs G’ = (V’, E’) with asso- ciated families of subsets R’ = {R: 1 i E I } , where R{ V’, 0 5 j 5 m, such that Go is a modification of G, G’” is constructed from G’, 0 i j 5 m - 1, and G” is the required graph G’. First, let Go be obtained from G by adding a new vertex r to V and, for each vertex x of G such that x E R, for some i E I , adding k edges from x to r. Let Ro = {Rp 1 i E I } where RP = R, U {r}. To construct G’” from G’, 0 I j 5 m - 1, assume, without loss of generality, that xJ E n:=l R, . We replace xf by a copy of the complete symmetric graph on z, new vertices vf,, i = 1 , . . . , tf . Next, each edge of G’ of the form (w,x , ) is replaced by a new vertex having a single edge to each vf, , 1 5 i I t’, a single edge from w, and k edges to r. Each edge of G’ of the form (xJ , w ’) is similarly replaced by a new vertex having a single edge from each v,, , 1 I i 5 t,, a sin-


gle edge to w ‘ , and k edges to r . Finally, let R ” ’ = {R;’’\ i E /}, where

ure 2 . Lemma 2 . I now follows from the following observations: RJ.1 , = (RI - x,) U {v,,}. This construction is illustratcd for 1, = k = 2 in Fig-

( i ) I f X is a family of vertices of V then Conjecture I . I ( 1) and (2) hold for X in G, for the family R, if and only if they hold for X in GI, for the family R’, Q s j s m .

(ii) Any R;’ for which r is the only common vertex with the other members of R1 in G’ may be replaced by a copy of { r } in Conjecture 1.1(1) and (2), O r j l m . I

To complete our reduction of Conjecture I . 1, it will be convenient to make a few further definitions:

2.2 Definitions. A directed graph G = ( V , E ) with a distinguished vertex r of outdegree zero will be denoted by the triple (V, E , r ) . If every vertex in V except r has outdegree precisely k, then G is called a k-ary graph (binary in the case k = 2). We say that G is edge (vertex) -linked to r if the condition of Conjecture 1.1’( 1) holds for r in G .

In proving that Conjecture 1 . 1 holds if it holds for k-ary graphs, we shall ap- peal to a result of Lovasz [4]. Thus, for a graph G = ( V , E , r) let cc;(x, r ) de- note the number of pairwise edge-disjoint paths from x to r in G .

2.3 Theorem (Lovbz). subgraph G ’ of G such that, for all vertices x E V - r ,

For any graph G = ( V , E , r ) , there is a spanning

cC.,(x, r ) = cc(x, r ) = outdegree of x in G‘.

2.4 Lemma. general.

If Conjecture I . 1 is true for k-ary graphs, then i t is true in

FIG. 2


Proof, By Lemma 2.1, it will be sufficient to show that Conjecture 1 . 1 ’ holds if i t holds for k-ary graphs. This will follow provided that, for any graph G = (V, E, r), if G is eage (vertex) -linked to r tnen G has a spanning ~ - q subgraph that is also edge (vertex) -linked to r. Let G ’ = (V’,E‘ , r ’ ) be obtained from G by adding a new vertex r ’ to V and adding k edges from r co r ’ . Then c J x , r ’ ) = k for all vertices x of V ‘ - r ’ , and Lemma 2.4 follows for edge-linkings by Theorem 2.3 and Menger’s Theorem. Now derive a graph G“ = ( V ” , E “ , r ‘ ) from G‘ as follows: for every vertex x of V ’ - r ‘ , add a ncw vertex .T’ and edge (s’ ,x); and for each edge of the form ( w , . ~ ) in E‘ . delete this edge and add a new edge (w, x ’). Clearly

k I

x E V” n V ’ x E V” - V ’ .

c&,r’) =

By Theorem 2 . 3 , G ” has a spanning subgraph H ” in which all vertices in V” fl V ’ have outdcgree k , all vertices in V” - V ’ have outdegree I . and cJx, r ’ ) = c,.(x, r ‘ ) = outdegree of .x in H ” for all .r E V” - r’ . But for any vertex .r E V, any k pairwise vertex-disjoint paths from x t o r in G can be placed in one to one correspondence with the k pairwise edge-disjoint paths from .x to r ’ in G”. Hence, Lemma 2.4 follows by Menger’s Theorem. I

3. PREDICATE FUNCTIONS

In this section we shall investigate B-vertex-linkings for the case k = 2. In order conveniently to manipulate these linkings we shall borrow a concept from computer science, whereby the vertices of outdegree 2 in binary graphs represent program tests or predicates, having a True and a False outcome represented by the two edges leaving the vertex (see [ 2 ] , for example). Each edge is accordingly labeled with a 1 or a 0, these values being treated arith- metically modulo 2.

3.1 Definitions. Let G = (V, E, r ) be a binary graph. A predicate function T

on G is a mapping of E to G F ( 2 ) , the binary Galois field, such that, lor each vertex .x E V with edges to y and y ’ , we have

(X , ) ’ )T f ( X , ! ’ ) T = 1 .

If P is a nontrivial path in G such that every edge is mapped by 7 to the same element of GF(2), then the image PT of P under 7 is defined to be this element. In this case, P is called 7-regular.

Let T be any predicate function on G = (V, E, r ) . We define a relation --i with respect to T on V by s -T y if and only if there exist vertex-disjoint w e g - ular paths P :x + r and P ‘ : y ---$ r such that PT + P’T = I . I f the relation -?

is reflexive, then clearly - 7 induces a pair of edge-disjoint r-branchings in G .


which satisfy the case k = 2 of Frank's conjecture (see Section 1) . We are in- terested in obtaining a ford relation on V (i.e., one for which x -r y for.all s and y), since this will immediately yield Conjecture 1 . 1 in the case k = 2. Thus we aim to prove:

3.2 Theorem. tion 7 on G for which -7 is total.

Let G be a binary graph. Then there exists a predicate func-

It will be convenient to refer to predicate functions that induce a reflexive or total relation on V as being themselves reflexive or total. The main step in proving Theorem 3.2 will be to establish that, in the case k = 2 , Frank's conjecture implies Conjecture I . I . In other words, we shall show that from a reflexive predicate function we can always construct a total predicate function. To achieve this, we shall require a method for deriving from an arbitrary predicate function 7, a new predicate function (T that is at least as good as 7 in the sense that any pair of vertices related under -7, is also related under -u.

3.3 Definition. Let T be any predicate function on a binary graph G = ( V , E , r ) and let P be a nontrivial 7-regular path to r in G such that PT = 0. De- fine U to be the set of all vertices u E V such that there exist a vertex u ' of P, u ' # r , and a nontrivial regular path P : u + u ' with PT = I . Define U * re- cursively by:

Rule (a): U * contains all vertices of P except r . Rule (b): If u E U has an edge ( u , u ) such that (u , u)7 = 0 and u E U * ,

Rule (c): No other vertices of V are in U *. then u E U * .

Then the extension of 7 with respect to P is the predicate function cr defined by, for all ( x , y ) E E:

where X " - ~ . : V * GF(2) is the characteristic function defined by

1 i f u E U - U * 0 i f u @ U - U * . X"-,,.(U) =

The following lemma says that cr-regular paths with cr = 0, starting in U * or U , respectively, do not exit from these sets (except possibly at r ) , and that (T-

regular paths, with (T = I , ending in U * or U , respectively, cannot have started outside these sets. From this we can deduce that regular paths in any extension of a predicate function are also regular in that predicate function, in the


case that such a path is located totally inside or totally outside U - U * (where the head of such a path is not taken into consideration).

3.4 Lemma. Let G , 7, P, U , U * , and u be as in Definition 3 . 3 . Let Po:po + p ; and PI : p , + p I be any nontrivial cr-regular paths in G such that P i u = i and pi’ f r , i = 0, 1 . Then

( 1 ) if u is a vertex of Po and u E U * , all vertices of P,:u + p ; are in U * . (2) if u is a vertex of PI and u E U * , all vertices of P , : p , + u are in U * .

( 1 ‘) if no vertices of Po are in U * and u is a vertex of Po and u E U - U *,

(2‘) if no vertices of P, are in U * and u is a vertex of f‘, and u E U - U *, all vertices of P,:u + p ; are in U - U * .

all vertices of PI :pI + u are in U - U *. Proof. We shall prove only ( I ) and (2) since ( 1 ’) and ( 2 ‘ ) are very similar. ( 1 ) : it E U * , so by Rules (a) and (b) of Definition 3 .3 , u must have a

r-regular path P to r in G with P r = 0 in which every vertex belongs to U * . Thus, if u has an edge to id‘ in P,,, it will be enough to show that (u, u’)r = 0, since then ( u , u ‘ ) must lie on P. Hence u ’ E U * , and we can repeat the argument for each vertex on P,:u + ph. But ( u , u’)cr = 0 and xu .u.(u) = 0 , so (u , u‘)r = 0 by definition of cr.

(2): Let ( u ’ , u) be an edge of PI. Then ( U ‘ , U ) C T = 1 , and if (u ‘ ,u ) r = 0 then u’ E U - U * C U . But then u’ E U * , by Rule (b) of Definition 3.3. Thus, we must have (u’ , u)r = 1. Then u’ E U by definition of U and the fact that u E U * implies u E U or u lies on P. Moreover, if u’ E U - U *, then (u ’ ,u )u = 1 # 0 = (u’,u)T + X ~ / - ~ . ( U ’ ) , Contradicting the definition of u. So u‘ E U * , and, similarly, all vertices of Pl:pl + u are in U * .

3.5 Proposition. Let T be a predicate function for a binary graph, G = (V, E , r ) , and let u be any extension of r. Then, for any x, v E V,x -? y implies x -u y.

Proof. Suppose for contradiction that, for some vertices p , q E V,p -7 4 but p +u q . Then there must exist vertices p ’. q ’ E V - r and u-regular paths P , : p + p ’ , P , : p + q ’ , Qo:q + q ’ , and Q , : q + p ’ , with at least one P, and at least one Q, nontrivial, and P,u = Q,u = i, i = 0, 1. Now let G ’ = (V’, E ‘ ) be the subgraph of G consisting of the sets E ‘ of all edges and V‘ of all vertices lying on the paths P, , Q , , i = 0, 1. Let 7’ = rIE. and u’ = ulL. be the restric- tions of r and u to G’. Now p --7 q and p f r r q , so r and u must disagree on some edge of G ‘ . Hence X~-~.(U) = 1 for some u E V’, where U and U * are defined as in Definition 3.3. However, suppose V’ n U * # $3. Then it follows from Lemma 3.4(1) and (2) that V ‘ fl U* = V‘. In particular, we must have u E U * , contradicting the assumption that x ~ , - ~ / . ( u ) = I . So V’ n U * = $3. But then V ‘ n U - U * # (l implies that V ’ n U - U * = V ’ by Lemma 3.4(1‘) and (2‘), whence 7’ = cr‘, i.e., T’ and cr’ agree on every edge of G ’ , again giving a contradiction. I


3.6 Theorem. predicate function. Then there exists a total predicate function on G.

Let G = (V, E , r ) be a binary graph and let G have a reflexive

Proof. Let 7 be a reflexive predicate function on G. If 7 is not total, then there exist distinct vertices p , q E V such that p -T p and y --T q but p +7 y. Then we can find nontrivial 7-regular paths P, : p + r and Q, : q + r , with the PI vertex-disjoint and the Q, vertex-disjoint and P,r = Q,T -= i, for i = 0, 1, and such that there exist vertices p ’, y ’ E V - r with p ’ lying in Po and Q , and y ’ lying in P , and Qo. Now construct the extension (T of 7 with respect to P(,. with U and U * defined with respect to P , as in Definition 3.3. We claim that p -v q, whence the result follows by taking successive extensions of 7 and applying Proposition 3.5. Now by definition of (T. it is clear that is identi- cally equal to T when restricted to the edges of Po and P , . Also by Proposition 3.5, we can find vertex-disjoint cT-regular paths Q , ‘ : q --$ r with Q,’a = i, i = 0, 1. If Q ; is vertex-disjoint from Po, then p --(, q as required. Conversely, suppose that Q ; meets Po at u , say, 14 # r . Then u E U * , so by Lemma 3.4(2), q E U * . Then by Lemma 3.4(1), every vertex of Q;I except r lies in U * . But the only vertex of PI in U * is p , and if QA passes p , then it must also pass u, and this is impossible since Q;I is vertex-disjoint from Q ;. So Q A and PI are vertex-disjoint, and again p -,, q. I

4. THE PROOF OF THE MAIN THEOREM

The proof of Theorem 3.2 will be by induction on the number of vertices of G. The inductive step is provided by the following operation on an edge (x.y) of G :

G(x, y ) denotes the graph obtained from G by deleting ( x , y) and identifying x with the head of the remaining edge having tail x (deleting the self-loop that arises).

We have

4.1 Lemma. Let G = (V, E , r) be a binary graph that is vertex-linked to r . Then for any vertex x E V - r of indegree at most 1, there is an edge (x,y) such that G ( x , y ) is vertex-linked to r.

Proof. If x has indegree zero , then for any edge ( x , y ) , we have G(x, y) = G - x . which is clearly vertex-linked to r. So suppose that x has indegree 1 , and let ( w , x ) , (w, w ’ ) , ( x , y , ) , and ( x , y 2 ) be edges of G. Suppose y,,y, # r, and define G ’ to be G with w identified with x (deleting the self- loop that arises). Then x has outdegree 3 in G ’ and G ’ is vertex-linked to r. We claim that this property is preserved in at least one of the graphs G ’ - ( x , y , ) = G ( x , y , ) , i = 1 ,2 , whence the result will follow. Indeed, suppose that deleting ( x , y , ) leaves some pair of vertices no longer vertex-linked to r, i = 1,2. By Menger’s Theorem, i t follows that there are partitions V = U, U


{c,} U V , , for some c, and i = 1,2, with .r and r in I / , and V , , respectively, and such that every path with tail in U , and head in V, must pass either the edge ( x , v, ) or the cut-vertex c,. But this contradicts the existence of vertex-disjoint paths P :x + r and Q : w ’ + r because we must have P n {cl, c2} # @ and {c,, c2} C Q. Finally, if y, = r, for i = I or 2, then clearly G(x, y,) is vertex- linked to r, f o r j # i . I

Proof of Theorem 3.2. The proof is by induction on n, the number of vertices of G. For n = 2, the result is clear. So let n 2 3 and suppose the result holds for all binary graphs with fewer vertices than G. Since G is binary, IEl = 21VI - 2 and there is some vertex p E V with indegree at most 1 . By Lemma 4.1, for some edge ( p , q o ) , G ( p , qo) is vertex-linked to I’. By inductive hypothesis, G ( p , q , ) has a total predicate function 7, and the theorem will follow from Theorem 3.6 if we can derive from 7 a reflexive predicate function 7’

on G. We can do this as follows: suppose the edges leaving p in G are ( p , q l ) . for i =- 0, I . Since 7 is total. there exist vertex-disjoint 7-regular paths P o : q , -+ I’ and P, : q , + r, where wc may assume, without loss of generality, that P,T = i, i = 0, I . Now if p has indegree zero in G, then G( p . q,,) = G - p , and we define 7’ to take the value z on ( p . q , ) , i = 0,1 and to be iden- tically equal to 7 on all other edges of G. Then 7‘ is clearly reflexive. Suppose, on the other hand, there is a (unique) vertex p ’ having an edge to p . Then ( p I , ql) is an edge of G ( p , q,), and 7’ is defined on E as follows:

If ( p ‘ , q , ) 7 = 1, then 7’ is clearly reflexive. If ( p ’ , q 1 ) 7 = 0 then, 7’ will be reflexive unless there exists a vertex u of G( p , 4,) such that the 7-regular vertex-disjoint paths Q, : u -+ r with Q,T = i , i = 0, 1 (which exist, since 7 is total) are such that Q, and Po are not vertex-disjoint and ( p ‘, q l ) lies on Q, in G ( p , q , ) . In this case, consider the paths P,’:q, -+ r having PL’7 = i + 1, i = 0, 1 . Again these paths exist because 7 is total; if they are vertex-disjoint, we just interchange 0 and 1 for the edges having tail p in the above definition of 7’ to obtain a reflexive predicate function. If they are not vertex-disjoint, then we deduce that u +7 qo, but this contradicts the hypothesis that 7 was total. This completes the proof.

ACKNOWLEDGMENTS

I originally proved Theorem 3.2 only for strongly connected graphs, relying on a much stronger condition on the graphs G(x,y). I am indebted to C. Hurkens (Tilburg) for showing me that this restriction was unnecessary by providing me


with Lemma 4.1, and for suggesting several other improvements. The use of Lovasz’s theorem to prove Lemma 2.4 was suggested to me by W. Jackson (London). Reference 6 was brought to my attention by A. Frank (Budapest).

References

[ 1 J J. Edmonds, Submodular functions, matroids and certain polyhedra. In Combinatorial Structures and Their Applications, R. Guy et al, Eds., Gordon and Breach, New York (1969) 69-87.

[ 2 ] N. E. Fenton, R. W. Whitty, and A. A. Kaposi, A generalized mathematical theory of structured programming. Theor. Comput. Sci. 36 (1985)

[3] A Frank, On the orientation of graphs. J . Cornbinat. Theory B 28 (1980)

[41 L. Lovasz, Connectivity in digraphs. J . Combinat. Theory B 15 (1973)

[5] C.St J. A. Nash-Williams, Well-balanced orientations of finite graphs and unobtrusive odd-vertex-pairings. In Recent Progress in Combinatorics, Academic Press, New York (1969) 133-149.

[6] A. Schrijver, Fractional packing and covering. In Packing and Covering in Combinatorics, A. Schrijver, Ed., Mathematisch Centrum, Amsterdam

[7] A. Schrijver, Min-max relations for directed graphs. In Bonn Workshop on Combinatorial Optimisation, Annals of Discrete Mathematics, Vol. 16, A. Bachem et al, Eds., North-Holland (1982) 261-280.

[8] P. Tong and E. L. Lawler, A faster algorithm for finding edge-disjoint branchings. Info. Process. Lett. 17(2) (1983) 73-76.

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Vertex-disjoint paths and edge-disjoint branchings in directed graphs