The competition-common enemy graph of a digraph
Discrete Applied Mathematics
Trapezoid graphs and their coloring
Discrete Applied Mathematics
The double competition number of some triangle-free graphs
Discrete Applied Mathematics
A special planar satisfiability problem and a consequence of its NP-completeness
Discrete Applied Mathematics
On the double competition number
Discrete Applied Mathematics
Competition Graphs of Hamiltonian Digraphs
SIAM Journal on Discrete Mathematics
Advances in Applied Mathematics
Linear-time transitive orientation
SODA '97 Proceedings of the eighth annual ACM-SIAM symposium on Discrete algorithms
On upper bound graphs with respect to operations on graphs
Theoretical Computer Science
On transformation of posets which have the same bound graph
Discrete Mathematics
On CCE graphs of doubly partial orders
Discrete Applied Mathematics
Journal of Combinatorial Theory Series B
Journal of Graph Theory
A class of acyclic digraphs with interval competition graphs
Discrete Applied Mathematics
The competition hypergraphs of doubly partial orders
Discrete Applied Mathematics
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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:@?w"1,w"2@?V(D),uw"1@?,vw"1@?,w"2u@?,w"2v@?@?A(D)}. A poset of dimension at most two is a digraph whose vertices are some points in the Euclidean plane R^2 and there is an arc going from a vertex (x"1,y"1) to a vertex (x"2,y"2) if and only if x"1x"2 and y"1y"2. 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 the doubly partial order competition number posed by Cho and Kim. We prove that the double competition graph of a poset of dimension at most two must be a trapezoid graph, generalizing a result of Kim, Kim, and Rho. Some connections are also established between the minimum numbers of isolated vertices required to be added to change a given graph into the competition graph, the double competition graph, of a poset and the minimum sizes of certain intersection representations of that graph.