On the 12-representability of induced subgraphs of a grid graph

TL;DR

This paper characterizes 12-representable square grid graphs using forbidden subgraphs, advances understanding of 12-representability, and explores related graph labeling questions, contributing to the broader classification of these graphs.

Contribution

It provides a complete forbidden subgraph characterization of 12-representable square grid graphs and proposes a conjecture for line grid graphs, advancing the theory significantly.

Findings

Complete characterization of square grid graphs via forbidden subgraphs

Conjecture and initial results on line grid graphs

Discussion of relations between labelings and 12-representability

Abstract

The notion of a 12-representable graph was introduced by Jones et al.. This notion generalizes the notions of the much studied permutation graphs and co-interval graphs. It is known that any 12-representable graph is a comparability graph, and also that a tree is 12-representable if and only if it is a double caterpillar. Moreover, Jones et al.\ initiated the study of 12-representability of induced subgraphs of a grid graph, and asked whether it is possible to characterize such graphs. This question in is meant to be about induced subgraphs of a grid graph that consist of squares, which we call square grid graphs. However, an induced subgraph in a grid graph does not have to contain entire squares, and we call such graphs line grid graphs. In this paper we answer the question of Jones et al.\ by providing a complete characterization of $12$-representable square grid graphs in terms of forbidden induced subgraphs. Moreover, we conjecture such a characterization for the line grid graphs and give a number of results towards solving this challenging conjecture. Our results are a major step in the direction of characterization of all 12-representable graphs since beyond our characterization, we also discuss relations between graph labelings and 12-representability, one of the key open questions in the area.