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.

翻译：设$G$为顶点集$V(G)$、边集$E(G)$的图。若存在字母表$V(G)$上的单词$w$，使得对任意两个不同字母$x,y \in V(G)$，字母$x$和$y$在$w$中交替出现当且仅当$(x,y) \in E(G)$，则称$G$为可词表示图。等价地，图可词表示当且仅当其具有半传递定向，即存在无环定向，使得对任意满足$m \ge 2$的有向路径$v_0 \rightarrow v_1 \rightarrow \cdots \rightarrow v_m$，要么$v_0$与$v_m$之间无弧，要么对所有$1 \le i < j \le m$，存在从$v_i$到$v_j$的弧。本文对可词表示共二部图（顶点集可划分为两个团的图类）提供全面的结构与算法刻画，统一了图论与矩阵论视角。首先证明共二部图为圆图当且仅当其为置换图，从而导出共二部圆图的最小禁止诱导子图刻画。核心贡献在于建立半传递性与二元矩阵的圆全相容性间的对应关系。除结构刻画外，本文利用Safe矩阵识别框架提出半传递共二部图的线性时间识别算法。