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

## TL;DR

This paper investigates the existence of arc-disjoint in- and out-branchings rooted at the same vertex in compositions of digraphs, providing conditions for strong and semicomplete cases and polynomial-time decision algorithms.

## Contribution

It generalizes previous results by characterizing when compositions of digraphs have good pairs at every vertex and offers polynomial-time algorithms for semicomplete compositions.

## Key findings

- Every strong digraph composition with each $n_i \\ge 2$ has a good pair at every vertex.
- Characterization of semicomplete compositions with a good pair, extending prior work.
- Polynomial-time decision procedure for the existence of a good pair in semicomplete compositions.

## 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 $u_1,\dots , u_t$ and let $H_1,\dots H_t$ be digraphs such that $H_i$ has vertices $u_{i,j_i},\ 1\le j_i\le n_i.$ Then the composition $Q=T[H_1,\dots , H_t]$ is a digraph with vertex set $\{u_{i,j_i}\mid 1\le i\le t, 1\le j_i\le n_i\}$ and arc set $$A(Q)=\cup^t_{i=1}A(H_i)\cup \{u_{ij_i}u_{pq_p}\mid u_iu_p\in A(T), 1\le j_i\le n_i, 1\le q_p\le n_p\}.$$ When $T$ is arbitrary, we obtain the following result: every strong digraph composition $Q$ in which $n_i\ge 2$ for every $1\leq i\leq t$, has a good pair at every vertex of $Q.$ The condition of $n_i\ge 2$ in this result cannot be relaxed. When $T$ is semicomplete, we characterize semicomplete compositions with a good pair, which generalizes the corresponding characterization by Bang-Jensen and Huang (J. Graph Theory, 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.