On word-representability of polyomino triangulations
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 $(x,y)$ is an edge in $E$. Some graphs are word-representable, others are not. It is known that a graph is word-representable if and only if it accep...
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| Main Authors: | , , |
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| Format: | Journal Article |
| Language: | English |
| Published: |
14-05-2014
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| Subjects: | |
| Online Access: | Get full text |
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| Summary: | 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
$(x,y)$ is an edge in $E$. Some graphs are word-representable, others are not.
It is known that a graph is word-representable if and only if it accepts a
so-called semi-transitive orientation.
The main result of this paper is showing that a triangulation of any convex
polyomino is word-representable if and only if it is 3-colorable. We
demonstrate that this statement is not true for an arbitrary polyomino. We also
show that the graph obtained by replacing each $4$-cycle in a polyomino by the
complete graph $K_4$ is word-representable. We employ semi-transitive
orientations to obtain our results. |
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| DOI: | 10.48550/arxiv.1405.3527 |