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- W4308758532 abstract "Jones et al. (2015) introduced the notion of $u$-representable graphs, where $u$ is a word over ${1, 2}$ different from $22cdots2$, as a generalization of word-representable graphs. Kitaev (2016) showed that if $u$ is of length at least 3, then every graph is $u$-representable. This indicates that there are only two nontrivial classes in the theory of $u$-representable graphs: 11-representable graphs, which correspond to word-representable graphs, and 12-representable graphs. This study deals with 12-representable graphs. Jones et al. (2015) provided a characterization of 12-representable trees in terms of forbidden induced subgraphs. Chen and Kitaev (2022) presented a forbidden induced subgraph characterization of a subclass of 12-representable grid graphs. This paper shows that a bipartite graph is 12-representable if and only if it is an interval containment bigraph. The equivalence gives us a forbidden induced subgraph characterization of 12-representable bipartite graphs since the list of minimal forbidden induced subgraphs is known for interval containment bigraphs. We then have a forbidden induced subgraph characterization for grid graphs, which solves an open problem of Chen and Kitaev (2022). The study also shows that a graph is 12-representable if and only if it is the complement of a simple-triangle graph. This equivalence indicates that a necessary condition for 12-representability presented by Jones et al. (2015) is also sufficient. Finally, we show from these equivalences that 12-representability can be determined in $O(n^2)$ time for bipartite graphs and in $O(n(bar{m}+n))$ time for arbitrary graphs, where $n$ and $bar{m}$ are the number of vertices and edges of the complement of the given graph." @default.
- W4308758532 created "2022-11-15" @default.
- W4308758532 creator A5076246585 @default.
- W4308758532 date "2022-11-09" @default.
- W4308758532 modified "2023-09-27" @default.
- W4308758532 title "Graph classes equivalent to 12-representable graphs" @default.
- W4308758532 doi "https://doi.org/10.48550/arxiv.2211.04871" @default.
- W4308758532 hasPublicationYear "2022" @default.
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