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

Let SS be any set of points in the Euclidean plane R2R2. For any p=(x,y)∈Sp=(x,y)∈S, put SW(p)={(x′,y′)∈S:x′xandy′>y}. Let GSGS be the graph with vertex set SS and edge set {pq:NE(p)∩NE(q)≠0̸andSW(p)∩SW(q)≠0̸}. We prove that the graph HH with V(H)={u,v,z,w,p,p1,p2,p3}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}E(H)={uv,vz,zw,wu,p1p3,p2p3,pu,pv,pz,pw,pp1,pp2,pp3} and the graph H′H′ obtained from HH by removing the edge pp3pp3 are both minimal forbidden subgraphs for the class of graphs of the form GSGS.