As Schnyder observes, the incidence poset of a graph "G" has order dimension two if and only if the graph is a path or a subgraph of a path. For, in when an incidence poset has order dimension is two, its only possible realizer consists of two total orders that (when restricted to the graph's vertices) are the reverse of each other. Any other two orders would have an intersection that includes an order relation between two vertices, which is not allowed for incidence posets. For these two orders on the vertices, an edge between consecutive vertices can be included in the ordering by placing it immediately following the later of the two edge endpoints, but no other edges can be included.