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.

翻译：图$G=(V,E)$被称为可词表示的，若存在一个字母表$V$上的词$w$，使得两个不同字母$x,y\in V$在$w$中交替出现当且仅当$xy\in E$。可词表示图构成一个被广泛研究的图类，与图定向、词组合学及图着色相关联。近三角剖分是指除了外部面之外的所有面均为三角形的平面图。此前已有若干近三角剖分子类在可词表示性框架下得到研究，包括多联骨牌三角剖分、带单个多米诺骨牌的矩形多联骨牌三角剖分、无$K_4$近三角剖分、三角网格图的面细分、网格覆盖圆柱图的三角剖分以及弦近三角剖分。本文通过禁止导出子图给出了可词表示近三角剖分的完整刻画。该结果统一并推广了上述子类已有的刻画结论，同时修正了先前工作中的不准确之处。