Forbidden Induced Subgraph Characterization of Word-Representable Co-bipartite Graphs

Eshwar Srinivasan; Ramesh Hariharasubramanian

Paper

Forbidden Induced Subgraph Characterization of Word-Representable Co-bipartite Graphs

Abstract

A graph

with vertex set

and edge set

is said to be word-representable if there exists a word

over the alphabet

such that, for any two distinct letters

, the letters

and

alternate in

if and only if

. Equivalently, a graph is word-representable if and only if it admits a semi-transitive orientation, that is, an acyclic orientation in which, for every directed path

with

, either there is no arc between

and

, or, for all

, there exists an arc from

. In this work, we provide a comprehensive structural and algorithmic characterization of word-representable co-bipartite graphs, a class of graphs whose vertex set can be partitioned into two cliques. This work unifies graph-theoretic and matrix-theoretic perspectives. We first establish that a co-bipartite graph is a circle graph if and only if it is a permutation graph, thereby deriving a minimal forbidden induced subgraph characterization for co-bipartite circle graphs. The central contribution then connects semi-transitivity with the circularly compatible ones property of binary matrices. In addition to the structural characterization, the paper introduces a linear-time recognition algorithm for semi-transitive co-bipartite graphs, utilizing Safe's matrix recognition framework.