Word-representable graphs, characterized by the existence of a semi-transitive orientation, form a well-studied class in graph theory. Comparability graphs form a subclass of word-representable graphs. Both classes are hereditary and hence admit characterizations in terms of forbidden induced subgraphs. While the minimal forbidden induced subgraphs for comparability graphs are known, a complete characterization for word-representable graphs remains open. In this paper, we investigate which minimal non-comparability graphs are also minimal non-word-representable graphs. To this end, we classify minimal non-comparability graphs according to whether they are word-representable or not, and thereby determine exactly which of them belong to the class of minimal non-word-representable graphs. As a consequence, we obtain a complete description of minimal non-word-representable graphs containing an all-adjacent vertex. We also consider an open problem posed by Kenkireth et al.\ concerning the cover number of word-representable graphs by comparability graphs. We show that there exist triangle-free word-representable graphs whose cover number by comparability graphs is at least $Ω(\log\log n)$. For triangle-free circle graphs, we prove that the cover number by comparability graphs is at most $3$, and that this bound is tight. More generally, we show that every circle graph $G$ with clique number $ω(G) \ge 24$ has cover number at most $2$. Finally, we identify four subclasses of word-representable graphs in which every graph has cover number at most $2$.

翻译：可词表示图（Word-representable graphs）可通过半传递定向的存在性刻画，是图论中一类被深入研究的图。可比图构成可词表示图的子类。这两类图均具有遗传性，因此可用禁止诱导子图进行刻画。尽管可比图的最小禁止诱导子图已知，但可词表示图的完整刻画仍是一个未解决问题。本文研究哪些最小非可比图同时也是最小非可词表示图。为此，我们根据最小非可比图是否可词表示对其进行分类，从而精确判定其中哪些属于最小非可词表示图的类。作为推论，我们得到包含全邻接顶点的最小非可词表示图的完整描述。此外，我们考虑了Kenkireth等人提出的关于可词表示图的可比图覆盖数的开放问题。我们证明存在三角形自由的可词表示图，其可比图覆盖数至少为$Ω(\log\log n)$。对于三角形自由的圆图，我们证明其可比图覆盖数至多为$3$，且该界是紧的。更一般地，我们证明每个团数$ω(G) \ge 24$的圆图$G$其覆盖数至多为$2$。最后，我们识别出可词表示图的四个子类，其中每个图的覆盖数至多为$2$。