Jones et al. (2015) introduced the notion of u-representable graphs, where u is a word over {1, 2} different from 22⋯2, 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(m̅+n)) time for arbitrary graphs, where n and m̅ are the number of vertices and edges of the complement of the given graph.
