Word-representability of subdivisions of triangular grid graphs

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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)∈ E. A triangular grid graph is a subgraph of a tiling of the plane with equilateral triangles defined by a finite number of triangles, called cells. A subdivision of a triangular grid graph is replacing some of its cells by plane copies of the complete graph K₄. Inspired by a recent elegant result of Akrobotu et al., who classified word-representable triangulations of grid graphs related to convex polyominoes, we characterize word-representable subdivisions of triangular grid graphs. A key role in the characterization is played by smart orientations introduced by us in this paper. As a corollary to our main result, we obtain that any subdivision of boundary triangles in the Sierpiński gasket graph is word-representable.