Independent branchings in acyclic diagraphs

Let D be a finite directed acyclic multigraph and t be a vertex of D such that for each other vertex x of D, there are n pairwise openly disjoint paths in D from x to t. It is proved that there exist n spanning trees BI , , B,, in D directed toward t such that for each vertex x # t of D, the N paths from x to t in B1, ,B,, are pairwise openly disjoint. @ 1999 Elsevier Science B.V. All rights reserved AMS classtfication: 05CO5; 05C40 Kqvwords: Branchings Throughout this paper, each digraph is finite and may have multiple edges but no loops. Digraphs without multiple edges directed equally are called simple. Let D be a digraph. V(D) and E(D) denote the set of the vertices of D and the set of the edges of D, respectively. For any distinct x,y E V(D), we denote by 6(D; x, y) the number of edges directed from x toward y. Moreover, we let 6+(D;x) and 6-(D;x) denote the number of edges directed from x and directed toward X, respectively. Paths and cycles in D are always directed and are not allowed to use a vertex more than once. A path from a vertex x to a vertex y is also called an X, y-path. As usual, digraphs without cycles are called acyclic. Let PI and Pl be two edge-disjoint paths in D. For each i E { 1,2}, let x, be the startvertex and y, be the endvertex of P,. Then PI and PZ are called disjoint if V(P,)flV(P*)= 0, nearly disjoint if V(PI)nV(P2)~{x,}n{x2}, and openly disjoint if V(P, ) n V(P2) C{x,, yl} n (~2, yl}. As usual, we call D n-(edge)-connected if for any two vertices x and y of D, there are at least n openly disjoint (edge-disjoint) X, y-paths in D. Moreover, if t is a vertex of D such that for all x E V(D)t, there are n pairwise openly disjoint (edge-disjoint) x, t-paths in D, then D is called (t, n)-(edge)-connected. ’ E-mail: [email protected]. 0012-365x199/$see front matter @ 1999 Elsevier Science B.V. All rights reserved PII: SOOl2-365X(98)00338-0 246 A. HucklDiscrete Mathematics 199 (1999) 245-249 An acyclic digraph B is called a branching if there exists t E V(B) with 6’(B; t) = 0 and 6’(B;x) = 1 for each x E V(B) t. t is called the root of B and B is also called a t-branching. For each x E V(B), xBt denotes the unique x, t-path in B. Now let BI and B2 be two branchings and for each i E { 1,2}, let ti be the root of Bi. Then B1 and BZ are called (weakly) independent if for each x E V(Bl )n V(B2), the paths XBI tl and xBzt2 are openly disjoint (edge-disjoint). A subdigraph D’ of a digraph D is called spanning if V(D’) = V(D). If D is a digraph and t E V(D), then an obviously necessary condition for the existence of n pairwise (weakly) independent spanning t-branchings in D is that D is (t,n)-(edge)connected. Ref. [2] proved that (t,n)-edge-connectivity is even sufficient for the existence of n pairwise weakly independent spanning t-branchings. So it is near to examine the following conjecture for each integer n 2 0. Conjecture 1. Let D be a (t,n)-connected digraph with t E V(D). Then there exist n pairwise independent spanning t-branchings in D. The following variation is due to A. Frank and appeared in [6]. Conjecture 1’. Let D be an n-connected digraph and t E V(D). Then there exist n pairwise independent spanning t-branchings in D. Clearly, for each integer n 20, Conjecture 1 implies Conjecture 1’. The converse also holds: Assume that D is a (t,n)-connected digraph with t E V(D). Clearly, we may assume that 6+(D; t) = 0. Let 6 := 6-(D; t) and let fl,. . . ,fs be the edges of D directed toward t. Moreover, take any n-connected digraph H with (V(H)/ 26 and any pairwise distinct vertices XI,. . . , x6 of H. Finally, let the digraph D’ be obtained from D by replacing t by H such that f i is directed toward xi in D’ for each i < 6 and by adding an edge directed from x toward y for each x E V(H) and y E V(D) t. It is standard to check that D’ is n-connected. Take any t’ E V(H). Then it is easy to see that any n pairwise independent spanning t’-branchings in D’ yield n pairwise independent spanning t-branchings in D. Clearly, Conjecture 1 and Conjecture 1’ are valid for n = 1 (and trivial for n = 0). [7] verified these conjectures for n = 2 (a short proof is given in [3]) and [4] constructed counterexamples for each n 23. Analogous statements for undirected graphs were proved for each n < 3 by [ 1,5] and are open for each n 2 4. With regard to the counterexamples of Conjectures 1 and l’, it is natural to look for additional conditions that ensure the existence of n pairwise independent spanning t-branchings. We will show that acyclicity is such a condition, i.e. we will prove the following theorem. Theorem 1. Let D be an acyclic (t, n)-connected digraph with t E V(D) and n 3 0. Then there exist n pairwise independent spanning t-branchings in D. A. Huckl Discrete Mathematics 199 ( 1999) 245-249 241 Note that there are non-trivial acyclic (t,n)-connected digraphs since we allow multiple edges (openly disjoint paths toward t may consist of parallel edges directed toward t). It turned out that it is more convenient to deal with modifications of Conjecture 1 and Theorem 1 where we have n pairwise distinct vertices tI,. . . t,, instead of a single vertex t. Let D be a digraph and tl,. . . , t,, E V(D) be pairwise distinct. If .X E V(D) and Pi is an X, t;-path in D for each i /O, Conjecture 1 and Conjecture 2 are equivalent (see also [3,4]). Particularly, Conjecture 2 is valid for all n 62 and false for all n 3 3. But we will show that this conjecture becomes valid for all n >, 0 if we restrict ourselves to acyclic digraphs. Obviously, to prove this restricted version, WC only have to consider simple digraphs and therefore it suffices to prove the following theorem. Theorem 2. Let D be a simple acyclic digraph and tl, . . . t,, E V(D) be pair&e; distinct with n 20. Moreover, assume that 6+(D;x)>,n @r each x E V(D) {tl.. . t,!}. Then there exists an independent (ti),,-system in D. Theorem 2 also implies Theorem 1: Assume that D is an acyclic (t, n)-connected digraph with t E V(D) and n 3 0. Obviously, we may assume that D t is simple. Let D’ be obtained from D-t by adding pairwise distinct new vertices tl , . . t,, and an edge directed from ,X toward ti for each x E V(D) t and each i E { 1,2, . . min(n, 6( D; x. t ) ) } Clearly, D’ is simple and acyclic. Moreover, since D is (t, n)-connected, it is easy to see that s+(D’;x) >n for each x E V(D) t. Thus by Theorem 2, there exists an independent (t,),,-system in D’. Using this system, it is easy to construct n pairwise independent spanning t-branchings in D. Let us prepare the proof of Theorem 2. It is well-known and easy to see that for each acyclic digraph D, there exists a numbering {.x! . . . ,x1 } of V(D) such that i > ,j for all i, j 0. Such a numbering is called topological. Now let D be a digraph and tl, . . . , t, be pairwise distinct vertices of D with n 3 0. Then D is 248 A. HucklDiscrete Mathematics 199 (1999) 245-249 called (ti),-admissible if the premises of Theorem 2 are satisfied, i.e. if D is acyclic and simple and if 6+(D; x) 2n for all x E V(D) {tl,. , . , 6,). Lemma 1. Let D be a (t;),-admissible digraph with pairwise distinct tl,. . , t,, E V(D) and n> 1. Then there exists a spanning t,,-branching B in D {tl,. . . , tn-l} and a topological numbering {xi ,. . ,x/} of V(B) tn such that D E(B) tn is (ti)n1 -admissible and {q,x/-I,..., xl} is a topological numbering of D-E(B)-{t ,,..., t,,}. Proof. We prove this lemma by induction on 1 V(D)/. Clearly, we may assume that V(D)-itI,...> tn} # 0. Take any topological numbering {y, , . . , yr} of D{tl , . . , tn}. Then clearly, also D * := D yr is (ti),-admissible. Thus by the induction-hypothesis, there exists a spanning &-branching B* of D* {tl, . ,&-I} and a topological numbering {xt,...,x/-1) of B* t,, such that D* E(B*) tn is (ti),-l-admissible and {X/-I,. . . ,x1 } is a topological numbering of D* -E(B*) {tl, . . . , t,,}. Define x0 := tn. Then since P(D; y/) an, we find a minimal j E (0,. . . , / 1) with 6(D; y/,x;) > 0. Let B be obtained from B* by adding y/ and the edge of D directed from y/ toward xi. Clearly, B is a spanning t,?-branching in D {t,, . . . , t,,1 }. Moreover, by construction, {.X1,. . . ,Xj, Y/,Xj+l). . . ) xl-1 } is a topological numbering of B tn and {X/-l~~~~~~j+l~Y/~~j~~~~~xl} is a topological numbering of D -E(B) {tl, . . , t,,}. Finally, it is easy to see that D E(B) t,, is (ti),-l-admissible (note that the unique edge of B directed from y/ is directed toward t,, if 6(D; y/, t,,) > 0). 0 Now we are able to prove Theorem 2. Let D be a (ti),-admissible digraph with pairwise distinct tl , . . . , t,, E V(D). We prove by induction on n that D contains an independent (ti),-system. Clearly, we may assume that n 3 1. Take a spanning &-branching B, of D {tl, . . . , &,-I} and a topological numbering {x1,. . .,x/} of B, tn according to Lemma 1. By the induction-hypothesis, there exists an independent (ti),-t-system (Bi),-l in D E(B,) t,,. Let j E { 1,. . . , r}. Then by construction, we have V(xjB,t,)C{t,,x,,...,xj} and V(xjBiti)~{ti.x/,...,xj} for all i<n 1. Thus (xjBiti)n is a (ti),-linkage and hence (Bi), is an independent (t,),-system. This completes the proof of Theorem 2. q Let us finally consider some algorithmic aspects. Following the proof of Lemma 1, we can develop an algorithm of complexity O(JE(D)() which constructs a t,,branching B and a topological numbering of B tn as described in Lemma 1 for each (ti),-admissible digraph D: Take any topological numbering (~1,. . . , yt} of D {t ,, . . .,t,}. Start with V(B) := {tn}, E(B) := 0, and the empty topological numbering of B t,,. Then add successively ~1,. . . , yr to B and to the topological numbering of B tn according to the proof of Lemma 1. Therefore, by the proof of Theorem 2, an independent (ti),-system in a (t,),-admissible graph D can be constructed in O(nIE(D)I) steps. Now by the derivation of Theorem 1 from Theorem 2, we see that A. Huckl Discrete Muthematics 199 (1999) 245-249 249 n pairwise independent spanning t-branchings in a (t,n)-connected acyclic digraph D can also be constructed in O(nlE(D)() steps.