On the complexity of recognizing Stick, BipHook and Max Point-Tolerance graphs

TL;DR

This paper proves that recognizing Stick, BipHook, and max point-tolerance graphs are all NP-complete problems, resolving several open questions in graph recognition complexity.

Contribution

It establishes the NP-completeness of recognizing Stick graphs and extends this result to BipHook and max point-tolerance graphs.

Findings

Recognizing Stick graphs is NP-complete.

Recognizing BipHook graphs is NP-complete.

Recognizing max point-tolerance graphs is NP-complete.

Abstract

Stick graphs are defined as follows. Let A (respectively B) be a set of vertical (respectively horizontal) segments in the plane such that the bottom endpoints of the segments in A and the left endpoints of the segments in B lie on the same ground straight line with slope -1. The Stick graph defined by A and B, which is necessarily bipartite, is the intersection graph of the segments in A with the segments in B. We answer an open problem by showing that recognizing Stick graphs is NP-complete. This result allows us to easily solve two other open problems, namely the recognition of BipHook graphs and of max point-tolerance graphs. We show that both of them are NP-complete problems.