On forbidden induced subgraphs for unit disk graphs

TL;DR

This paper investigates forbidden induced subgraphs for unit disk graphs, revealing infinitely many new minimal non-unit disk graphs and exploring structural properties of co-bipartite unit disk graphs.

Contribution

It systematically studies forbidden induced subgraphs for unit disk graphs, introducing new structural tools and characterizing certain co-bipartite subclasses.

Findings

Identified infinitely many new minimal non-unit disk graphs.

Provided structural characterization of specific co-bipartite unit disk graphs.

Proposed a conjecture on the closure of co-bipartite unit disk graphs under bipartite complementation.

Abstract

A unit disk graph is the intersection graph of disks of equal radii in the plane. The class of unit disk graphs is hereditary, and therefore admits a characterization in terms of minimal forbidden induced subgraphs. In spite of quite active study of unit disk graphs very little is known about minimal forbidden induced subgraphs for this class. We found only finitely many minimal non unit disk graphs in the literature. In this paper we study in a systematic way forbidden induced subgraphs for the class of unit disk graphs. We develop several structural and geometrical tools, and use them to reveal infinitely many new minimal non unit disk graphs. Further we use these results to investigate structure of co-bipartite unit disk graphs. In particular, we give structural characterization of those co-bipartite unit disk graphs whose edges between parts form a C4-free bipartite graph, and show that bipartite complements of these graphs are also unit disk graphs. Our results lead us to propose a conjecture that the class of co-bipartite unit disk graphs is closed under bipartite complementation.