Characterization of Word-Representable Graphs using Modular Decomposition
Tithi Dwary, K. V. Krishna
TL;DR
This paper characterizes word-representable graphs through modular decomposition by showing that replacing a module $M$ at a vertex with $G_a[M]$ preserves word-representability exactly when the base graph $G$ is word-representable and the module $M$ is a comparability graph, with a tight formula for the permutation-representation number $\mathcal{R}^p$. Leveraging this, it provides a complete modular-decomposition-based criterion: a decomposable graph $G$ is word-representable iff every module $G[M_i]$ is a comparability graph and the quotient $G/\mathcal{P}$ is word-representable, and it gives the explicit representation-number bound $\mathcal{R}(G)=\max\{\mathcal{R}(G/\mathcal{P}), \mathcal{R}^p(G[M_1]),\ldots, \mathcal{R}^p(G[M_k])\}$. The paper also resolves the open problem on the word-representability of the lexicographical product $G[G']$, showing $G[G']$ is word-representable iff $G$ is word-representable and $G'$ is a comparability graph, with $\mathcal{R}(G[G'])=\max\{\mathcal{R}(G), \mathcal{R}^p(G')\}$. Collectively, these results yield a polynomial-time test for non-word-representability via modular decomposition and offer a pathway to identify new classes of word-representable graphs through structural decomposition.
Abstract
In this work, we characterize the class of word-representable graphs with respect to the modular decomposition. Consequently, we determine the representation number of a word-representable graph in terms of the permutation-representation numbers of the modules and the representation number of the associated quotient graph. In this connection, we also obtain a complete answer to the open problem posed by Kitaev and Lozin on the word-representability of the lexicographical product of graphs.