Strong arc decompositions of split digraphs

Abstract

A strong arc decomposition of a digraph is a partition of its arc set into two sets such that the digraph is strong for . Bang‐Jensen and Yeo conjectured that there is some such that every ‐arc‐strong digraph has a strong arc decomposition. They also proved that with one exception on four vertices every 2‐arc‐strong semicomplete digraph has a strong arc decomposition. Bang‐Jensen and Huang 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 split digraph is a digraph whose underlying undirected graph is a split graph, meaning that its vertices can be partitioned into a clique and an independent set. Equivalently, a split digraph is any digraph which can be obtained from a semicomplete digraph by adding a new set of vertices and some arcs between and . 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.