由态射生成的分裂图的词表示性
Word-representability of split graphs generated by morphisms
摘要：
图$G=(V,E)$是词可表示的，当且仅当在字母表$V$上存在词$w$，使得字母$x$和$y$，$x\neq y$在$w$中交替，当且仅当$xy\in E$。分裂图是一个顶点可以划分为一个团和一个独立集的图。关于词可表示图的研究已有很长的历史，最近，分裂图的词可表示性问题引起了人们的兴趣。本文首先利用邻接矩阵的列置换给出了词可表示分裂图的一个刻画。然后，我们专注于研究分裂图的词表示性，分裂图是由一个态射的迭代得到的，这个概念来自于词的组合学。我们证明了一些一般定理，并提供了一个完整的分类的情况下，态射定义的$2\times 2$矩阵。
英文摘要：
A graph $G=(V,E)$ is word-representable if and only if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$, $x\neq y$, alternate in $w$ if and only if $xy\in E$. A split graph is a graph in which the vertices can be partitioned into a clique and an independent set. There is a long line of research on word-representable graphs in the literature, and recently, word-representability of split graphs has attracted interest. In this paper, we first give a characterization of word-representable split graphs in terms of permutations of columns of the adjacency matrices. Then, we focus on the study of word-representability of split graphs obtained by iterations of a morphism, the notion coming from combinatorics on words. We prove a number of general theorems and provide a complete classification in the case of morphisms defined by $2\times 2$ matrices.
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