Editing Preorder (section)

===Graph theory===

* The [[reachability]] relationship in any [[directed graph]] (possibly containing cycles) gives rise to a preorder, where <math>x \lesssim y</math> in the preorder if and only if there is a path from ''x'' to ''y'' in the directed graph. Conversely, every preorder is the reachability relationship of a directed graph (for instance, the graph that has an edge from ''x'' to ''y'' for every pair {{nowrap|(''x'', ''y'')}} with <math>x \lesssim y</math>). However, many different graphs may have the same reachability preorder as each other. In the same way, reachability of [[directed acyclic graph]]s, directed graphs with no cycles, gives rise to [[partially ordered set]]s (preorders satisfying an additional antisymmetry property).
* The [[graph-minor]] relation is also a preorder.