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

Joanna N. Chen, Sergey Kitaev

Research output: Contribution to journalArticlepeer-review

## Abstract

The notion of a 12-representable graph was introduced by Jones, Kitaev, Pyatkin and Remmel in [Representing graphs via pattern avoiding words, Electron. J. Combin. 22 (2015) #P2.53]. 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 of Jones et al. 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.

Original language English 383-403 21 Discussiones Mathematicae Graph Theory 42 2 9 Dec 2019 https://doi.org/10.7151/dmgt.2263 Published - 31 Mar 2022

## Keywords

• graph representation
• 12-representable graph
• grid graph
• forbidden subgraph
• square grid graph
• line grid graph

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