What Makes the Recognition Problem Hard for Classes Related to Segment and String graphs?

TL;DR

This paper investigates the computational complexity of recognizing certain intersection graph classes, demonstrating NP-hardness even under restrictions like large girth and bounded degree, and providing structural insights for unit grid intersection graphs.

Contribution

It establishes NP-hardness of recognition for UGIGs, GIGs, and string graphs under various restrictions, and offers structural results for UGIG representations.

Findings

Recognition of these classes is NP-hard even with large girth.

Recognition remains hard for graphs with degree restrictions (4, 5, 8).

Structural bounds on UGIG representations are provided.

Abstract

We explore what could make recognition of particular intersection-defined classes hard. We focus mainly on unit grid intersection graphs (UGIGs), i.e., intersection graphs of unit-length axis-aligned segments and grid intersection graphs (GIGs, which are defined like UGIGs without unit-length restriction) and string graphs, intersection graphs of arc-connected curves in a plane. We show that the explored graph classes are NP-hard to recognized even when restricted on graphs with arbitrarily large girth, i.e., length of a shortest cycle. As well, we show that the recognition of these classes remains hard even for graphs with restricted degree (4, 5 and 8 depending on a particular class). For UGIGs we present structural results on the size of a possible representation, too.