The double competition multigraph of a digraph

In this article, we introduce the notion of the double competition multigraph of a digraph. We give characterizations of the double competition multigraphs of arbitrary digraphs, loopless digraphs, reflexive digraphs, and acyclic digraphs in terms of edge clique partitions of the multigraphs.


Introduction
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). The competition 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 N + D (x) ∩ N + D (y) = ∅. This notion 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 [10] 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 [4,5,9,11] for recent results on double competition graphs.
A  [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 m xy multiple edges between two distinct vertices x and y, where m xy is the nonnegative integer defined by m xy = |N + D (x) ∩ N + D (y)|. See [8,12] for recent results on competition multigraphs.
In this article, we introduce the notion of the double competition multigraph of a digraph, and we give characterizations of the double competition multigraphs of arbitrary digraphs, loopless digraphs, reflexive digraphs, and acyclic digraphs in terms of edge clique partitions of the multigraphs.

Main Results
We define the double competition multigraph of a digraph as follows.
Definition. Let D be a digraph. The double competition multigraph of D is the multigraph which has the same vertex set as D and has m xy multiple edges between two distinct vertices x and y, where m xy is the nonnegative integer defined by where A i and B j are the sets defined by Proof. First, we show the only-if part. Let M be the double competition multigraph of an arbitrary digraph D. Let (v 1 , . . . , v n ) be an ordering of the vertices of D. For i, j ∈ [n], we define Then S ij is a clique of M. Let F be the family of S ij 's whose size is at least two, i.e., By the definition of a double competition multigraph, F is an edge clique partition of M.
We show that the condition (I) holds. Fix i and j in [n] and let A i and B j be sets as defined in (1) and (2).
There are four cases for v k arising from the definitions of A i and B j as follows: Hence the condition (I) holds.
Next, we show the if part. Let M be a multigraph with n vertices, and suppose that there exists an ordering (v 1 , . . . , v n ) of the vertices of M and a double indexed edge clique partition F = {S ij | i, j ∈ [n]} of M such that the condition (I) holds.
We define a digraph D by V (D) := V (M) and . Again, take any two distinct vertices v k and v l and let t ′ := m M ′ ({v k , v l }). Then, for some nonnegative integers r ′ and s ′ with r ′ s ′ = t ′ , there are r ′ common in-neighbors v i 1 , . . . , v i r ′ and s ′ common out-neighbors v j 1 , . . . , v j s ′ of the vertices v k and v l in D.
Therefore, {v k , v l } ⊆ A i ∩B j for any i ∈ {i 1 , . . . , i r ′ } and any j ∈ {j 1 , . . . , j s ′ }. By the condition (I), we have A i ∩ B j = S ij . Therefore {v k , v l } ⊆ S ij for any i ∈ {i 1 , . . . , i r ′ } and any j ∈ {j 1 , . . . , j s ′ } and this implies that where A i and B j are the sets defined as (1) and (2).
Proof. First, we show the only-if part. Let M be the double competition multigraph of a loopless digraph D. Let (v 1 , . . . , v n ) be an ordering of the vertices of D. Let S ij (i, j ∈ [n]) be the sets defined as (3), and let F be the family defined as (4). Then S ij is a clique of M, and F is an edge clique partition of M. Moreover, we can show, as in the proof of Theorem 1, that the condition (I) holds. Now we show that the condition (II) holds. Take any vertex v k ∈ S ij . Then (v i , v k ), (v k , v j ) ∈ A(D). Since D is loopless, we have v i = v k and v i = v k . Therefore it follows that v i ∈ S ij and v j ∈ S ij . Thus the condition (II) holds. Next where A i , B j , S i * , and S * i are the sets defined as (1) and (2).
Proof. First, we show the only-if part. Let M be the double competition multigraph of a reflexive digraph D. Let (v 1 , . . . , v n ) be an ordering of the vertices of D. Let S ij (i, j ∈ [n]) be the sets defined as (3), and let F be the family defined as (4). Then S ij is a clique of M, and F is an edge clique partition of M. Moreover, we can show, as in the proof of Theorem 1, that the condition (I) holds. Now we show that the condition (III) holds. Since D is reflexive, we have (v i , v i ) ∈ A(D) for any i ∈ [n]. Then it follows from the definition of D that there exists p ∈ [n] such that v i ∈ S ip or v i ∈ S pi . Therefore v i ∈ S i * ∪ S * i . Thus the condition (III) holds.
Next, we show the if part. Let M be a multigraph with n vertices, and suppose that there exists an ordering (v 1 , . . . , v n ) of the vertices of M and a double indexed edge clique partition F = {S ij | i, j ∈ [n]} of M such that the conditions (I) and (III) hold. We define a digraph D by V (D) := V (M) and A(D) given in (5). Fix any i ∈ [n]. By the condition (III), there exists p ∈ [n] such that v i ∈ S ip or v i ∈ S pi . Then it follows from the definition of D that (v i , v i ) ∈ A(D). Therefore D is a reflexive digraph. Moreover we can show, as in the proof of Theorem 1, that M is the double competition multigraph of D.
A digraph D is said to be acyclic if D has no directed cycles. An ordering (v 1 , . . . , v n ) of the vertices of a digraph D, where n is the number of vertices of D, is called an acyclic ordering of D if (v i , v j ) ∈ A(D) implies i < j. It is well known that a digraph D is acyclic if and only if D has an acyclic ordering. (IV) for any i, j, k ∈ [n], v k ∈ S ij implies i < k < j, where A i and B j are the sets defined as (1) and (2).
Proof. First, we show the only-if part. Let M be the double competition multigraph of an acyclic digraph D. Let (v 1 , . . . , v n ) be an acyclic ordering of the vertices of D. Let S ij (i, j ∈ [n]) be the sets defined as (3), and let F be the family defined as (4). Then S ij is a clique of M, and F is an edge