Characterization of Word-Representable Graphs using Modular Decomposition

TL;DR

This paper characterizes word-representable graphs through modular decomposition, linking their representation number to modules and quotient graphs, and resolves an open problem on lexicographical products.

Contribution

It introduces a modular decomposition-based characterization of word-representable graphs and solves an open problem regarding their lexicographical products.

Findings

Representation number expressed via modules and quotient graphs

Complete solution to the open problem on lexicographical product

Characterization of word-representable graphs using modular 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.