Decomposing and colouring some locally semicomplete digraphs

TL;DR

This paper investigates subclasses of locally semicomplete digraphs, providing structural decompositions and applications such as partitioning vertices into acyclic digraphs under certain conditions.

Contribution

It offers new structural decomposition theorems for subclasses of locally semicomplete digraphs and resolves a conjecture related to vertex partitioning into acyclic digraphs.

Findings

Structural decomposition theorems for subclasses of locally semicomplete digraphs

Partitioning vertices into two acyclic digraphs under specific conditions

Resolution of a conjecture by Naserasr and others

Abstract

A digraph is semicomplete if any two vertices are connected by at least one arc and is locally semicomplete if the out-neighbourhood and the in-neighbourhood of any vertex induce a semicomplete digraph. In this paper we study various subclasses of locally semicomplete digraphs for which we give structural decomposition theorems. As a consequence we obtain several applications, among which an answer to a conjecture of Naserasr and the first and third authors: if an oriented graph is such that the out-neighbourhood of every vertex induces a transitive tournament, then one can partition its vertex set into two acyclic digraphs.