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  "body": "\n## Definition\n\nLet $(P, \\le)$ be a [[partially ordered set|0QV2JS9RJ3]].\nThe *comparability graph* of $P$ is the [[graph|00C0BR7D7H]] with vertex set $P$ in which distinct $x, y \\in P$ are adjacent if and only if $x \\le y$ or $y \\le x$.\nA graph $G$ is a *comparability graph* if it is isomorphic to the comparability graph of some partially ordered set.\n\n## Remarks\n\nEquivalently, $G$ is a comparability graph when its edges can be oriented so that $x \\to y$ and $y \\to z$ always imply $x \\to z$; such an orientation is called transitive.\nExample: the complete bipartite graph $K_{m,n}$ is the comparability graph of the poset with $m$ minimal elements, $n$ maximal elements, and each minimal element below each maximal element.\nNon-example: the cycle $C_5$ is not a comparability graph.\nIts largest clique has $2$ vertices but it needs $3$ colours, while in a comparability graph the clique number equals the chromatic number, by [[Mirsky's theorem|5VPXR3RDF1]].\nThe cliques of the comparability graph are the chains of $P$, and its independent sets are the antichains of $P$.\nDifferent partial orders, such as an order and its dual, can have the same comparability graph.\n",
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