Word-representable graphs, characterized by the existence of a semi-transitive orientation, form a well-studied class of graphs. Comparability graphs form another well-studied class and constitute a subclass of word-representable graphs. Both classes are hereditary and admit characterizations in terms of minimal forbidden induced subgraphs. While the minimal forbidden induced subgraphs for comparability graphs are completely characterized, the corresponding characterization for word-representable graphs remains open. In this paper, we precisely determine which minimal non-comparability graphs are also minimal non-word-representable graphs by classifying minimal non-comparability graphs according to whether they are word-representable. As a consequence, we provide a complete description of minimal non-word-representable graphs containing an all-adjacent vertex. We also address an open problem posed by Kenkireth et al.\ concerning the cover number of word-representable graphs by comparability graphs. We demonstrate the existence of word-representable graphs on $n$ vertices whose cover number by comparability graphs is $Ω(\log n)$, which establishes that the universal $O(\log n)$ upper bound is asymptotically tight for the class of word-representable graphs. For triangle-free circle graphs, we establish that the cover number by comparability graphs is at most $3$ and demonstrate that this bound is tight. More generally, we prove that for any circle graph $G$ with clique number $ω(G) \ge 24$, the cover number by comparability graphs is at most $2$. Finally, we identify four subclasses of word-representable graphs for which the cover number by comparability graphs of every graph in these classes is at most $2$.

翻译：词可表示图由半传递定向的存在性刻画，是经过深入研究的一类图。可比图构成另一类被充分研究的图，且是词可表示图的子类。这两类图均具有遗传性，并可通过极小禁止导出子图加以刻画。尽管可比图的极小禁止导出子图已被完全刻画，但词可表示图的相应刻画问题仍未解决。本文通过将极小非可比图依据其是否具有词可表示性进行分类，精确确定了哪些极小非可比图同时也是极小非词可表示图。据此，我们给出了包含全邻接点的极小非词可表示图的完整描述。此外，我们解决了Kenkireth等人提出的关于可比图覆盖词可表示图的覆盖数这一开放问题。我们证明了存在$n$个顶点上且被可比图覆盖的覆盖数为$Ω(\log n)$的词可表示图，这表明通用上界$O(\log n)$对于词可表示图类而言是渐近紧的。对于无三角形圆图，我们证明其可比图覆盖数至多为$3$，并说明该界是紧的。更一般地，我们证明对于任意团数$ω(G) \ge 24$的圆图$G$，其可比图覆盖数至多为$2$。最后，我们识别出词可表示图的四个子类，其中每个图的可比图覆盖数至多为$2$。