Obstructions to chordal circular-arc graphs of small independence number

TL;DR

This paper investigates the structural properties of chordal graphs related to blocking quadruples and establishes conditions under which such graphs are circular-arc, providing forbidden subgraph characterizations and a geometric construction method.

Contribution

It introduces a forbidden subgraph characterization for chordal graphs without blocking quadruples and shows that absence of blocking quadruples ensures circular-arc graphs among chordal graphs with small independence number.

Findings

Chordal graphs without blocking quadruples are characterized by specific forbidden subgraphs.

Absence of blocking quadruples guarantees circular-arc structure in chordal graphs with independence number less than five.

A novel geometric method constructs circular-arc representations using clique trees.

Abstract

A blocking quadruple (BQ) is a quadruple of vertices of a graph such that any two vertices of the quadruple either miss (have no neighbours on) some path connecting the remaining two vertices of the quadruple, or are connected by some path missed by the remaining two vertices. This is akin to the notion of asteroidal triple used in the classical characterization of interval graphs by Lekkerkerker and Boland. We show that a circular-arc graph cannot have a blocking quadruple. We also observe that the absence of blocking quadruples is not in general sufficient to guarantee that a graph is a circular-arc graph. Nonetheless, it can be shown to be sufficient for some special classes of graphs, such as those investigated by Bonomo et al. In this note, we focus on chordal graphs, and study the relationship between the structure of chordal graphs and the presence/absence of blocking quadruples. Our contribution is two-fold. Firstly, we provide a forbidden induced subgraph characterization of chordal graphs without blocking quadruples. In particular, we observe that all the forbidden subgraphs are variants of the subgraphs forbidden for interval graphs. Secondly, we show that the absence of blocking quadruples is sufficient to guarantee that a chordal graph with no independent set of size five is a circular-arc graph. In our proof we use a novel geometric approach, constructing a circular-arc representation by traversing around a carefully chosen clique tree.