等价于12-可表示图的图类
Graph classes equivalent to 12-representable graphs
摘要：
Jones等人（2015）引入了$u$ -可表示图的概念，其中$u$是$\{1, 2\}$上不同于$22\cdots2$的词，作为词可表示图的推广。Kitaev（2016）证明了如果$u$的长度至少为3，则每个图都是$u$ -可表示的。这表明在$u$ -可表示图的理论中只有两类非平凡图：11-可表示图和12-可表示图，它们对应于词可表示图。本研究涉及12-可表示图。Jones et al.（2015）提供了12-可表示树在禁止诱导子图方面的特征。Chen和Kitaev（2022）给出了12-可表示格图的一个子类的禁止诱导子图刻画。本文证明了一个二部图是12-可表示的当且仅当它是一个区间包含二部图。由于区间包含偶图的最小禁止诱导子图的列表是已知的，因此等价性给出了12-可表示偶图的禁止诱导子图的刻画。然后，我们有一个禁止诱导子图的格图的特征，它解决了陈和Kitaev（2022）的一个公开问题。研究还表明，一个图是12-可表示的当且仅当它是一个简单三角形图的补图。这种等价性表明Jones等人（2015）提出的12-可表示性的必要条件也是充分的。最后，我们从这些等价性，12-可表示性可以确定在$O(n^2)$时间的二部图和在$O(n(\bar{m}+n))$时间的任意图，其中$n$和$\bar{m}$是给定的图的补的顶点和边的数目。
英文摘要：
Jones et al. (2015) introduced the notion of $u$-representable graphs, where $u$ is a word over $\{1, 2\}$ different from $22\cdots2$, 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.
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