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 $2$.

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