6 research outputs found
Graph classes equivalent to 12-representable graphs
Jones et al. (2015) introduced the notion of -representable graphs, where
is a word over different from , as a generalization
of word-representable graphs. Kitaev (2016) showed that if is of length at
least 3, then every graph is -representable. This indicates that there are
only two nontrivial classes in the theory of -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 time for bipartite graphs and
in time for arbitrary graphs, where and are the
number of vertices and edges of the complement of the given graph.Comment: 12 pages, 6 figure