Poset , competition numbers , and interval graph ∗

Let D = (V (D), A(D)) be a digraph. The competition graph of D, is the graph with vertex set V (D) and edge set {uv ∈ `V (D) 2 ́ : ∃w ∈ V (D),−→ uw,−→ vw ∈ A(D)}. The double competition graph of D, is the graph with vertex set V (D) and edge set {uv ∈ `V (D) 2 ́ : ∃w1, w2 ∈ V (D),−−→ uw1,−−→ vw1,−−→ w2u,−−→ w2v ∈ A(D)}. A poset of dimension at most two is a digraph whose vertices are some points in the Euclidean plane R and there is an arc going from a vertex (x1, y1) to a vertex (x2, y2) if and only if x1 > x2 and y1 > y2. We show that a graph is the competition graph of a poset of dimension at most two if and only if it is an interval graph at least half of whose maximal cliques are isolated vertices. This answers an open question on doubly partial order competition number posed in [H.H. Cho, S-R. Kim, A class of acyclic digraphs with interval competition graphs, Discrete Appl. Math. 148 (2005), 171–180.] We prove that the double competition graph of a poset of dimension at most two must be a trapezoid graph, generalizing the main results of [S-J. Kim, S-R. Kim, Y. Rho, On CCE graphs of doubly partial orders, Discrete Appl. Math. 155 (2007), 971–978.]. It is also established in this note some connections between the minimum numbers of isolated vertices required to be added to make a given graph into the competition graph, respectively, double competition graph, of a poset and the minimum sizes of certain intersection representations of that graph.