A co-bipartite graph is the complement of a bipartite graph, equivalently a graph whose vertices can be partitioned into two disjoint cliques. Word-representability requires a word over the vertex set in which two distinct letters alternate exactly when the corresponding vertices are adjacent. The paper notes that both word-representable and non-word-representable co-bipartite graphs are known, including non-word-representable examples such as the complements of the graphs T_1 and T_2.
The paper studies several subclasses, proving word-representability for complements of even and odd paths, even cycles, generalized crown graphs, and co-bipartite graphs with one clique of size two, while also deriving conditions for semi-transitive orientations in co-bipartite graphs. These results provide partial progress toward a complete characterization, but the general class remains unresolved.