Two minimal forbidden subgraphs for double competition graphs of posets of dimension at most two

Let S be any set of points in the Euclidean plane R2. For any p = (x, y) ∈ S, put SW (p) = {(x, y) ∈ S : x < x and y < y} and NE(p) = {(x, y) ∈ S : x > x and y > y}. Let GS be the graph with vertex set S and edge set {pq : NE(p) ∩ NE(q) 6= ∅ and SW (p) ∩ SW (q) 6= ∅}. We prove that the graphH with V (H) = {u, v, z, w, p, p1, p2, p3} and E(H) = {uv, vz, zw, wu, p1p3, p2p3, pu, pv, pz, pw, pp1, pp2, pp3} and the graph H ′ obtained from H by removing the edge pp3 are both minimal forbidden subgraphs for the class of graphs of the form GS . © 2008 Elsevier Ltd. All rights reserved.