Forbidden patterns of graphs 12-representable by pattern-avoiding words

TL;DR

This paper explores graphs representable by pattern-avoiding words, characterizing forbidden patterns for certain classes and linking them to well-known graph classes like trivially perfect and bipartite permutation graphs.

Contribution

It provides forbidden pattern characterizations for 12-representable graphs represented by words avoiding specific patterns, extending understanding of their structure and classification.

Findings

Graphs avoiding pattern 111 are trivially perfect.

Graphs avoiding pattern 121 are trivially perfect.

Graphs avoiding pattern 231 are bipartite permutation graphs.

Abstract

A graph $G = (\{1, 2, \ldots, n\}, E)$ is $12$-representable if there is a word $w$ over $\{1, 2, \ldots, n\}$ such that two vertices $i$ and $j$ with $i < j$ are adjacent if and only if every $j$ occurs before every $i$ in $w$. These graphs have been shown to be equivalent to the complements of simple-triangle graphs. This equivalence provides a characterization in terms of forbidden patterns in vertex orderings as well as a polynomial-time recognition algorithm. The class of $12$-representable graphs was introduced by Jones et al. (2015) as a variant of word-representable graphs. A general research direction for word-representable graphs suggested by Kitaev and Lozin (2015) is to study graphs representable by some specific types of words. For instance, Gao, Kitaev, and Zhang (2017) and Mandelshtam (2019) investigated word-representable graphs represented by pattern-avoiding words. Following this research direction, this paper studies $12$-representable graphs represented by words that avoid a pattern $p$. Such graphs are trivial when $p$ is of length $2$. When $p = 111$, $121$, $231$, and $321$, the classes of such graphs are equivalent to well-known classes, such as trivially perfect graphs and bipartite permutation graphs. For the cases where $p = 123$, $132$, and $211$, this paper provides forbidden pattern characterizations.