Arc-disjoint in- and out-branchings rooted at the same vertex in compositions of digraphs

Abstract

A digraph D=(V,A) has a good pair at a vertex r if D has a pair of arc-disjoint in- and out-branchings rooted at r. Let T be a digraph with t vertices u1,…,ut and let H1,…Ht be digraphs such that Hi has vertices ui,ji,1≤ji≤ni. Then the composition Q=T[H1,…,Ht] is a digraph with vertex set {ui,ji∣1≤i≤t,1≤ji≤ni} and arc set A(Q)=∪i=1tA(Hi)∪{uijiupqp∣uiup∈A(T),1≤ji≤ni,1≤qp≤np}. If T is strongly connected, then Q is called a strong composition and if T is semicomplete, i.e., there is at least one arc between every pair of vertices, then Q is called a semicomplete composition.We obtain the following result: every strong digraph composition Q in which ni≥2 for every 1≤i≤t, has a good pair at every vertex of Q. The condition of ni≥2 in this result cannot be relaxed. We characterize semicomplete compositions with a good pair, which generalizes the corresponding characterization by Bang-Jensen and Huang (1995) for quasi-transitive digraphs. As a result, we can decide in polynomial time whether a given semicomplete composition has a good pair rooted at a given vertex.