**Incidence Posets of Graphs**

The incidence poset of any undirected graph *G* has the vertices and edges of *G* as its elements; in this poset, *x* ≤ *y* if either *x* = *y* or *x* is a vertex, *y* is an edge, and *x* is an endpoint of *y*. Certain kinds of graphs may be characterized by the order dimensions of their incidence posets: a graph is a path graph if and only if the order dimension of its incidence poset is at most two, and according to Schnyder's theorem it is a planar graph if and only if the order dimension of its incidence poset is at most three (Schnyder 1989).

Read more about this topic: Order Dimension

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