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

Arc-disjoint strong spanning subdigraphs in compositions and products of digraphs

A digraph $D=(V,A)$ has a good decomposition if $A$ has two disjoint sets $A_1$ and $A_2$ such that both $(V,A_1)$ and $(V,A_2)$ 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 $\{u_{i,j_i}\mid 1\le i\le t, 1\le j_i\le n_i\}$ and arc set $$A(Q)=\cup^t_{i=1}A(H_i)\cup \{u_{ij_i}u_{pq_p}\mid u_iu_p\in A(T), 1\le j_i\le n_i, 1\le q_p\le n_p\}.$$ For digraph compositions $Q=T[H_1,\dots H_t]$, we obtain sufficient conditions for $Q$ to have a good decomposition and a characterization of $Q$ with a good decomposition when $T$ is a strong semicomplete digraph and each $H_i$ is an arbitrary digraph with at least two vertices. For digraph products, we prove the following: (a) if $k\geq 2$ is an integer and $G$ is a strong digraph which has a collection of arc-disjoint cycles covering all vertices, then the Cartesian product digraph $G^{\square k}$ (the $k$th powers with respect to Cartesian product) has a good decomposition; (b) for any strong digraphs $G, H$, the strong product $G\boxtimes H$ has a good decomposition

Packing Strong Subgraph in Digraphs

In this paper, we study two types of strong subgraph packing problems in digraphs, including internally disjoint strong subgraph packing problem and arc-disjoint strong subgraph packing problem. These problems can be viewed as generalizations of the famous Steiner tree packing problem and are closely related to the strong arc decomposition problem. We first prove the NP-completeness for the internally disjoint strong subgraph packing problem restricted to symmetric digraphs and Eulerian digraphs. Then we get inapproximability results for the arc-disjoint strong subgraph packing problem and the internally disjoint strong subgraph packing problem. Finally we study the arc-disjoint strong subgraph packing problem restricted to digraph compositions and obtain some algorithmic results by utilizing the structural properties

Strong arc decompositions of split digraphs

A {\bf strong arc decomposition} of a digraph $D=(V,A)$ is a partition of its arc set $A$ into two sets $A_1,A_2$ such that the digraph $D_i=(V,A_i)$ is strong for $i=1,2$. Bang-Jensen and Yeo (2004) conjectured that there is some $K$ such that every $K$-arc-strong digraph has a strong arc decomposition. They also proved that with one exception on 4 vertices every 2-arc-strong semicomplete digraph has a strong arc decomposition. Bang-Jensen and Huang (2010) extended this result to locally semicomplete digraphs by proving that every 2-arc-strong locally semicomplete digraph which is not the square of an even cycle has a strong arc decomposition. This implies that every 3-arc-strong locally semicomplete digraph has a strong arc decomposition. A {\bf split digraph} is a digraph whose underlying undirected graph is a split graph, meaning that its vertices can be partioned into a clique and an independent set. Equivalently, a split digraph is any digraph which can be obtained from a semicomplete digraph $D=(V,A)$ by adding a new set $V'$ of vertices and some arcs between $V'$ and $V$. In this paper we prove that every 3-arc-strong split digraph has a strong arc decomposition which can be found in polynomial time and we provide infinite classes of 2-strong split digraphs with no strong arc decomposition. We also pose a number of open problems on split digraphs

Arc-disjoint out- and in-branchings in compositions of digraphs

An out-branching $B^+_u$ (in-branching $B^-_u$) in a digraph $D$ is a connected spanning subdigraph of $D$ in which every vertex except the vertex $u$, called the root, has in-degree (out-degree) one. A {\bf good $\mathbf{(u,v)}$-pair} in $D$ is a pair of branchings $B^+_u,B^-_v$ which have no arc in common. Thomassen proved that is NP-complete to decide if a digraph has any good pair. A digraph is {\bf semicomplete} if it has no pair of non adjacent vertices. A {\bf semicomplete composition} is any digraph $D$ which is obtained from a semicomplete digraph $S$ by substituting an arbitrary digraph $H_x$ for each vertex $x$ of $S$. Recently the authors of this paper gave a complete classification of semicomplete digraphs which have a good $(u,v)$-pair, where $u,v$ are prescribed vertices of $D$. They also gave a polynomial algorithm which for a given semicomplete digraph $D$ and vertices $u,v$ of $D$, either produces a good $(u,v)$-pair in $D$ or a certificate that $D$ has such pair. In this paper we show how to use the result for semicomplete digraphs to completely solve the problem of deciding whether a given semicomplete composition $D$, has a good $(u,v)$-pair for given vertices $u,v$ of $D$. Our solution implies that the problem is polynomially solvable for all semicomplete compositions. In particular our result implies that there is a polynomial algorithm for deciding whether a given quasi-transitive digraph $D$ has a good $(u,v)$-pair for given vertices $u,v$ of $D$. This confirms a conjecture of Bang-Jensen and Gutin from 1998