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 \((x,y)\in E\). 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 face subdivision of a triangular grid graph is replacing some of its cells by plane copies of the complete graph \(K_4\). 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 face 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 face subdivision of boundary triangles in the Sierpiński gasket graph is word-representable.
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Notes
Not all planar graphs are word-representable. The minimum non-word-representable (planar) graph is the wheel graph \(W_5\) obtained by the face subdivision (in our sense) of the cycle graph \(C_5\).
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Acknowledgments
The authors are grateful to the two anonymous referees for their useful suggestions. This work was supported by the 973 Project, the PCSIRT Project of the Ministry of Education and the National Science Foundation of China. Also, the second author is thankful to the Center for Combinatorics at Nankai University for its hospitality during the author’s stay in November 2014.
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Chen, H.Z.Q., Kitaev, S. & Sun, B.Y. Word-Representability of Face Subdivisions of Triangular Grid Graphs. Graphs and Combinatorics 32, 1749–1761 (2016). https://doi.org/10.1007/s00373-016-1693-z
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DOI: https://doi.org/10.1007/s00373-016-1693-z