Stick graphs: examples and counter-examples

TL;DR

This paper explores the properties of Stick graphs, providing new examples, non-examples, and certificates for recognition, and investigates their relationship with other graph classes like circle and circular arc graphs.

Contribution

It proves that complements of circle and circular arc graphs are Stick graphs, introduces certificates for non-Stick graphs, and examines their hierarchy with chordal grid intersection graphs.

Findings

Complements of circle graphs are Stick graphs.

Certificates for non-Stick graphs are proposed.

All known non-Stick graphs contain long holes.

Abstract

Grid intersection graphs are the intersection graphs of vertical and horizontal segments in the plane. When the bottom and respectively left endpoints of the vertical and horizontals segments belong to a line with negative slope, the graph is called a Stick graph. Very few results exist on Stick graphs: only small classes of Stick graphs have been identified; recognizing Stick graphs is an open problem; and even building examples of graphs that are not Stick graphs is quite tricky. In this paper, we first prove that the complements of circle graphs and of circular arc graphs are Stick graphs. Then, we propose two certificates allowing to decide that a graph is not a Stick graph, and use them to build new examples of non-Stick graphs. It turns out that these examples of non-Stick graphs, as well as all those from literature, have long holes. We thus also investigate the place of chordal grid intersection graphs in the hierarchy of classes built around Stick graphs.