On semi-transitive orientability of split graphs

TL;DR

This paper proves that recognizing semi-transitive orientability of split graphs can be done in polynomial time and characterizes such graphs with small independent sets, extending previous classifications.

Contribution

It establishes a polynomial-time recognition algorithm for semi-transitive split graphs and characterizes those with small independent sets via minimal forbidden subgraphs.

Findings

Recognition of semi-transitive split graphs is polynomial-time solvable.

Characterization of semi-transitive split graphs with independent set size ≤ 3.

Extended classification of semi-transitive split graphs with small clique sizes.

Abstract

A directed graph is semi-transitive if and only if it is acyclic and for any directed path $u_1\rightarrow u_2\rightarrow \cdots \rightarrow u_t$, $t \geq 2$, either there is no edge from $u_1$ to $u_t$ or all edges $u_i\rightarrow u_j$ exist for $1 \leq i < j \leq t$. Recognizing semi-transitive orientability of a graph is an NP-complete problem. A split graph is a graph in which the vertices can be partitioned into a clique and an independent set. Semi-transitive orientability of spit graphs was recently studied in the literature. The main result in this paper is proving that recognition of semi-transitive orientability of split graphs can be done in a polynomial time. We also characterize, in terms of minimal forbidden induced subgraphs, semi-transitively orientable split graphs with the size of the independent set at most 3, hence extending the known classification of such graphs with the size of the clique at most 5.