and bi in P. And conversely, any family of linear orders that includes one order of this type for each i has P as its intersection. Thus, P has dimension exactly n. In fact, P is known as the standard example of a poset of dimension n, and is usually denoted by Sn. Order dimension two The partial orders with order dimension two may be characterized as the partial orders whose comparability graph is the complement of the comparability graph of a different partial order . That is, P is a partial order with order dimension two if and only if there exists a partial order Q on the same set of elements, such that every pair x, y of distinct elements is comparable in exactly one of these two partial orders. If P is realized by two linear extensions, then partial order Q complementary to P may be realized by reversing one of the two linear extensions. Therefore, the comparability graphs of the partial orders of dimension two are exactly the permutation graphs, graphs that are both themselves comparability graphs and complementary to comparability graphs. The partial orders of order dimension two include the series-parallel partial orders . They are exactly the partial orders whose Hasse diagrams have dominance drawings, which can be obtained by using the positions in the two permutations of a realizer as Cartesian coordinates. Computational complexity It is possible to determine in polynomial time whether a given finite partially ordered set has order dimension at most two, for instance, by testing whether the comparability graph of the partial order is a permutation graph. However, for any k ≥ 3, it is NP-complete to test whether the order dimension is at most k . 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 . For