On 132-representable Graphs

TL;DR

This paper introduces the concept of 132-representable graphs, exploring their properties, showing they are circle graphs, and providing explicit representations for small graphs and complete graphs.

Contribution

It merges the theories of word-representable graphs and pattern-avoiding words, establishing 132-representability as a new graph class and characterizing its properties.

Findings

Every 132-representable graph is a circle graph.

Trees and cycle graphs are 132-representable.

Explicit 132-avoiding representations are provided for small graphs and complete graphs.

Abstract

A graph $G = (V,E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $w$ if and only if $xy$ is an edge in $E$. Word-representable graphs are the subject of a long research line in the literature initiated in \cite{KP}, and they are the main focus in the recently published book \cite{KL}. A word $w=w_1\cdots w_{n}$ avoids the pattern $132$ if there are no $1\leq i_1<i_2<i_3\leq n$ such that $w_{i_1}<w_{i_3}<w_{i_2}$. The theory of patterns in words and permutations is a fast growing area discussed in \cite{HM,Kit}. A research direction suggested in \cite{KL} is in merging the theories of word-representable graphs and patterns in words. Namely, given a class of pattern-avoiding words, can we describe the class of graphs represented by the words? Our paper provides the first non-trivial results in this direction. We say that a graph is 132-representable if it can be represented by a 132-avoiding word. We show that each 132-representable graph is necessarily a circle graph. Also, we show that any tree and any cycle graph are 132-representable, which is a rather surprising fact taking into account that most of these graphs are non-representable in the sense specified, as a generalization of the notion of a word-representable graph, in \cite{JKPR}. Finally, we provide explicit 132-avoiding representations for all graphs on at most five vertices, and also describe all such representations, and enumerate them, for complete graphs.