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

Abstract

A graph $G$ with vertex set $V(G)$ and edge set $E(G)$ is said to be word-representable if there exists a word $w$ over the alphabet $V(G)$ such that, for any two distinct letters $x,y \in V(G)$, the letters $x$ and $y$ alternate in $w$ if and only if $(x,y) \in E(G)$. 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 $v_0 \rightarrow v_1 \rightarrow \cdots \rightarrow v_m$ with $m \ge 2$, either there is no arc between $v_0$ and $v_m$, or, for all $1 \le i < j \le m$, there exists an arc from $v_i$ to $v_j$. 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.