On word-representability of polyomino triangulations

TL;DR

This paper characterizes when triangulations of convex polyominoes are word-representable, showing they are if and only if 3-colorable, and explores related properties using semi-transitive orientations.

Contribution

It establishes a precise condition for word-representability of convex polyomino triangulations and extends understanding of graph transformations affecting word-representability.

Findings

Convex polyomino triangulations are word-representable iff they are 3-colorable.

Not all polyomino triangulations are word-representable.

Replacing 4-cycles with K4 graphs preserves word-representability.

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 $(x,y)$ is an edge in $E$. Some graphs are word-representable, others are not. It is known that a graph is word-representable if and only if it accepts a so-called semi-transitive orientation. The main result of this paper is showing that a triangulation of any convex polyomino is word-representable if and only if it is 3-colorable. We demonstrate that this statement is not true for an arbitrary polyomino. We also show that the graph obtained by replacing each $4$-cycle in a polyomino by the complete graph $K_4$ is word-representable. We employ semi-transitive orientations to obtain our results.