Word-representability of triangulations of grid-covered cylinder graphs

TL;DR

This paper characterizes the word-representability of triangulations of grid-covered cylinder graphs, showing it depends on avoiding specific subgraphs, extending previous results on grid graphs to cylindrical structures.

Contribution

It provides a new characterization of word-representability for triangulations of grid-covered cylinder graphs based on forbidden subgraphs, generalizing prior work on grid graphs.

Findings

Word-representability is characterized by avoiding six minimal induced subgraphs for three-sector cases.

For more than three sectors, avoiding wheel graphs W5 and W7 determines word-representability.

The results extend the understanding of word-representability from grid graphs to cylindrical structures.

Abstract

A graph $G=(V,E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$, $x\neq y$, alternate in $w$ if and only if $(x,y)\in E$. Halld\'{o}rsson et al.\ have shown that a graph is word-representable if and only if it admits a so-called semi-transitive orientation. A corollary to this result is that any 3-colorable graph is word-representable. Akrobotu et al.\ have shown that a triangulation of a grid graph is word-representable if and only if it is 3-colorable. This result does not hold for triangulations of grid-covered cylinder graphs, namely, there are such word-representable graphs with chromatic number 4. In this paper we show that word-representability of triangulations of grid-covered cylinder graphs with three sectors (resp., more than three sectors) is characterized by avoiding a certain set of six minimal induced subgraphs (resp., wheel graphs $W_5$ and $W_7$).