Word-representability of triangulations of rectangular polyomino with a single domino tile

TL;DR

This paper extends a known characterization of word-representability from convex polyomino triangulations to those with a single domino tile, broadening the understanding of polyomino properties.

Contribution

It generalizes the result that triangulations are word-representable if and only if 3-colorable to include polyominoes with one domino tile.

Findings

Triangulations with one domino tile are word-representable if and only if they are 3-colorable.

The existing characterization for convex polyominoes extends to certain non-convex cases.

The result broadens the class of polyominoes for which word-representability can be characterized.

Abstract

A recent elegant result of Akrobotu et al. states that a triangulation of any convex polyomino is word-representable if and only if it is 3-colorable. In this paper, we generalize a particular case of this result by showing that the result of Akrobotu et al. is true even if we allow a domino tile, instead of having just $1\times 1$ tiles on a rectangular polyomino.