Strong arc decompositions of split digraphs

A strong arc decomposition of a digraph $D=\left(V,A\right)$ is a partition of its arc set $A$ into two sets $A_1,A_2$ such that the digraph $D_i=\left(V,A_i\right)$ is strong for $i=1,2$. Bang-Jensen and Yeo 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 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 $D=\left(V,A\right)$ by adding a new set $V^{\prime }$ of vertices and some arcs between $V^{\prime }$ 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.