The double competition multigraph of a digraph

The competition graph of a digraph is defined to be the intersection graph of the family of the out-neighborhoods of the vertices of the digraph (see [6] for intersection graphs). A digraph D is a pair (V (D), A(D)) of a set V (D) of vertices and a set A(D) of ordered pairs of vertices, called arcs. An arc of the form (v, v) is called a loop. For a vertex x in a digraph D, we denote the out-neighborhood of x in D by N D (x) and the in-neighborhood of x in D by N− D (x), i.e., N + D (x) := {v ∈ V (D) | (x, v) ∈ A(D)} and N− D (x) := {v ∈ V (D) | (v, x) ∈ A(D)}. A graph is a pair (V (G), E(G)) of a set V (G) of vertices and a set E(G) of unordered pairs of vertices, called edges. The competition graph of a digraphD is the graph which has the same vertex set asD and has an edge between two distinct vertices x and y if and only if N D (x)∩N + D (y) ̸= ∅. R. D. Dutton and R. C. Brigham [3] and F. S. Roberts and J. E. Steif [8] gave characterizations of competition graphs by using edge clique covers of graphs. The notion of competition graphs was introduced by J. E. Cohen [2] in 1968 in connection with a problem in ecology, and several variants and generalizations of competition graphs have been studied. In 1987, D. D. Scott [11] introduced the notion of double competition graphs as a variant of the notion of competition graphs. The double competition graph (or the competition-common enemy graph or the CCE graph) of a digraph D is the graph which has the same vertex set as D and has an edge between two distinct vertices x and y if and only if both N D (x) ∩N + D (y) ̸= ∅ and N − D (x) ∩N − D (y) ̸= ∅ hold. See [4, 5, 10, 12] for recent results on double competition graphs. A multigraph M is a pair (V (M), E(M)) of a set V (M) of vertices and a multiset E(M) of unordered pairs of vertices, called edges. Note that, in our definition, multigraphs have no loops. We may consider a multigraph M as the pair (V (M),mM) of the vertex set V (M) and the nonnegative integer-valued function mM : ( V 2 ) → Z≥0 on the set ( V 2 ) of all unordered pairs of V where mM({x, y}) is defined to be the number of multiple edges between the vertices x and y in M . The notion of competition multigraphs was introduced by C. A. Anderson, K. F. Jones, J. R. Lundgren, and T. A. McKee [1] in 1990 as a variant of the notion of competition graphs. The competition multigraph of a digraph D is the multigraph which has the same vertex set as D and has mxy multiple edges between two distinct vertices x and y, where mxy is the nonnegative integer defined by mxy = |N D (x) ∩ N + D (y)|. See [9, 13] for recent results on competition multigraphs. In this talk, we introduce the notion of the double competition multigraph of a