Characterization of Word-Representable Near-Triangulations

A graph $G=(V,E)$ is said to be word-representable if there exists a word $w$ over the alphabet $V$ such that two distinct letters $x,y\in V$ alternate in $w$ if and only if $xy \in E$. Word-representable graphs form a well-studied graph class with connections to graph orientations, combinatorics on words, and graph coloring. A near-triangulation is a planar graph in which every face except the outer face is a triangle. Several subclasses of near-triangulations have previously been investigated in the context of word-representability, including polyomino triangulations, triangulations of rectangular polyominoes with a single domino tile, $K_4$-free near-triangulations, face subdivisions of triangular grid graphs, triangulations of grid-covered cylinder graphs, and chordal near-triangulations. In this paper, we obtain a complete characterization of word-representable near-triangulations in terms of forbidden induced subgraphs. Our result unifies and extends the previously known characterizations for the above subclasses, while also correcting inaccuracies appearing in earlier works.