4 research outputs found
Semi-strict chordality of digraphs
Chordal graphs are important in algorithmic graph theory. Chordal digraphs
are a digraph analogue of chordal graphs and have been a subject of active
studies recently. In this paper we introduce the notion of semi-strict chordal
digraphs which form a class strictly between chordal digraphs and chordal
graphs. We characterize semi-strict chordal digraphs by forbidden subdigraphs
within the cases of locally semicomplete digraphs and weakly quasi-transitive
digraphs.Comment: 12 pages, 2 figures. arXiv admin note: text overlap with
arXiv:2008.0356
Strong arc decompositions of split digraphs
A {\bf 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 (2004) conjectured that there is some
such that every -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 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