Arc-disjoint Strong Spanning Subdigraphs of Semicomplete Compositions

A strong arc decomposition of a digraph $D=(V,A)$ is a decomposition of its arc set $A$ into two disjoint subsets $A_1$ and $A_2$ such that both of the spanning subdigraphs $D_1=(V,A_1)$ and $D_2=(V,A_2)$ are strong. 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 $\cup_{i=1}^t V(H_i)=\{u_{i,j_i}\mid 1\le i\le t, 1\le j_i\le n_i\}$ and arc set \[ \left(\cup^t_{i=1}A(H_i) \right) \cup \left( \cup_{u_iu_p\in A(T)} \{u_{ij_i}u_{pq_p} \mid 1\le j_i\le n_i, 1\le q_p\le n_p\} \right). \] We obtain a characterization of digraph compositions $Q=T[H_1,\dots H_t]$ which have a strong arc decomposition when $T$ is a semicomplete digraph and each $H_i$ is an arbitrary digraph. Our characterization generalizes a characterization by Bang-Jensen and Yeo (2003) of semicomplete digraphs with a strong arc decomposition and solves an open problem by Sun, Gutin and Ai (2018) on strong arc decompositions of digraph compositions $Q=T[H_1,\dots , H_t]$ in which $T$ is semicomplete and each $H_i$ is arbitrary. Our proofs are constructive and imply the existence of a polynomial algorithm for constructing a \good{} decomposition of a digraph $Q=T[H_1,\dots , H_t]$, with $T$ semicomplete, whenever such a decomposition exists.