The competition number of a generalized line graph is at most two

In 1982, Opsut showed that the competition number of a line graph is at most two and gave a necessary and sufﬁcient condition for the competition number of a line graph being one. In this paper, we generalize this result to the competition numbers of generalized line graphs, that is, we show that the competition number of a generalized line graph is at most two, and give necessary conditions and sufﬁcient conditions for the competition number of a generalized line graph being one.


Introduction
The notion of a competition graph was introduced by Cohen [1] as a means of determining the smallest dimension of ecological phase space.The competition graph C(D) of a digraph D is a graph which has the same vertex set as D and an edge between two distinct vertices u and v if and only if there exists a vertex x in D such that (u, x) and (v, x) are arcs of D. Roberts [12] observed that any graph G together with sufficiently many isolated vertices is the competition graph of an acyclic digraph.The competition number k(G) of a graph G is defined to be the smallest nonnegative integer k such that G together with k isolated vertices added is the competition graph of an acyclic digraph.It is not easy in general to compute k(G) for an arbitrary graph G, since Opsut [8] showed that the computation of the competition number of a graph is an NP-hard problem.It has been one of the important research problems in the study of competition graphs to compute the exact values of the competition numbers of various graphs.For some special graph families, we have explicit formulae for computing competition numbers: If G is a chordal graph without isolated vertices, then k(G) = 1; If G is a nontrivial triangle-free connected graph then ([12]).For more recent results on graphs whose competition numbers are calculated exactly, see [3,4,5,6,7,9,10,11,13].
The line graph L(H) of a graph H is the simple graph defined by V (L(H)) = E(H) and ee ′ ∈ E(L(H)) if and only if e and e ′ have a vertex in common and e = e ′ .A graph G is called a line graph if there exists a graph H such that G is isomorphic to the line graph of H.A clique S of a graph G is a set of vertices of G such that the subgraph induced by S is a complete graph (the empty set is also considered a clique).A vertex v in a graph G is called simplicial if the neighborhood of v in G is a clique of G.In 1982, Opsut [8] showed the following theorem.In this paper, we investigate the competition number of a generalized line graph which was introduced by Hoffman [2] in 1970.For a positive integer m, the cocktail party graph CP (m) is the complete multipartite graph with m partite sets all of which have the size two: where v ∈ V (H) with f (v) > 0 by adding edges between all the vertices in Q v and e ∈ V (L(H)) such that e is incident to v in H (see Figure 1).For a graph H, if f is a zero function (i.e.f (v) = 0 for any v ∈ V (H)), then the generalized line graph of (H, f ) is the same as the line graph of H.A graph G is called a generalized line graph if there exists a vertex-weighted graph (H, f ) such that G is isomorphic to the generalized line graph of (H, f ).
In this paper, we show the following result.

Theorem 1.2 The competition number of a generalized line graph is at most two.
This paper is organized as follows.Section 2 is the main part of this paper.Subsection 2.1 gives some observations on the competition graphs of acyclic digraphs which will be used in this paper.Subsection 2.2 shows that the competition number of a generalized line graph is at most two.In Subsection 2.3, we investigate generalized line graphs whose competition numbers are one, and give some sufficient conditions and necessary conditions.Section 3 gives some concluding remarks.

Preliminaries
For a digraph D and a vertex v of D, N + D (v) and N − D (v) denotes the out-neighborhood and the inneighborhood of v, respectively, i.e., It is well-known that a digraph is acyclic if and only if it has an acyclic ordering.
For a clique S of a graph G and an edge e of G, we say that e is covered by S if both of the endvertices of e are contained in S.An edge clique cover of a graph G is a family of cliques of G such that each edge of G is covered by some clique in the family.The edge clique cover number θ E (G) of a graph G is the minimum size of an edge clique cover of G.A vertex clique cover of a graph G is a family of cliques of G such that each vertex of G is contained in some clique in the family.The vertex clique cover number θ V (G) of a graph G is the minimum size of a vertex clique cover of G.For a graph G and a vertex v of G, θ V (N G (v)) is the vertex clique cover number of the subgraph of G induced by the neighbors of v in G. Opsut [8] showed the following lower bound for the competition number of a graph (see also [14] for a generalization of this inequality).

Proposition 2.1 ([8]) For any graph
For a positive integer k, we denote by I k the edgeless graph on k vertices, i.e., the graph with k vertices and no edges.The following lemma which comes from an easy observation is elementary but useful.

Lemma 2.2 Let G be a graph with at least two vertices and let k be an integer such that k ≥ k(G). Then there exists an acyclic digraph
By the definition of an acyclic ordering of a digraph, it holds that Then D is an acyclic digraph satisfying (a), (b), and (c).Thus the statement holds.✷ The competition number of a cocktail party graph is given by Kim, Park and Sano [6].

Proposition 2.3 ([6])
The competition number of a cocktail party graph CP (m) with m ≥ 2 is equal to two.

Proof of Theorem 1.2
In this subsection, we show that the competition number of a generalized line graph is at most two.
For two vertex-disjoint graphs G and H and a clique K of G, we define the graph G ⋉ K H by (see Figure 2).From Lemma 2.2, we introduce a notion of top-two of a graph as follows.Proof: Suppose that H has no edges.Let V (H) = {u 1 , u 2 , . . ., u m }, where m ≥ 2. We define a digraph where u m+1 and u m+2 are new vertices.Then, it is easy to see that D is an acyclic digraph satisfying (i) and (ii).
Suppose that H has no isolated vertices.Let k := k(H).By the assumption that {u 1 , u 2 } is a top-two of H, there exists an acyclic digraph D ′′ such that C(D ′′ ) = H ∪ I k and the two vertices u 1 and u 2 in Proof: Let G := L(H, f ) for convenience.First, we consider the case where f is a zero function.We show the theorem by induction on the number of edges of H.If H has at most one edge, then the statement is checked easily.Assume that the statement is true for any graph with m − 1 edges where m ≥ 2. Let H be a graph with m edges.It is sufficient to consider the case where H is connected.Take an edge e = uv of H. Since m ≥ 2 and H is connected, there exists an edge e ′ incident to e in H. Without loss of generality, we may assume that the vertex u is also an endvertex of e ′ .Let H ′ be the graph obtained from H by deleting the edge e.Then L(H ′ ) is the graph obtained from L(H) by deleting the vertex e.Since H ′ has m − 1 edges, by the induction hypothesis, there exists an acyclic digraph Then the ordering of the vertices in D obtained from an acyclic ordering of D ′ by adding the vertex z 2 to it as the last term is an acyclic ordering of D, and so D is acyclic.By the definitions of D and H ′ , Thus the theorem holds when f is a zero function.
Second, we consider the case where f is not a zero function.Let v 1 , v 2 , . . ., v n be the vertices of H such that f (v i ) > 0. For each i ∈ {0} ∪ [n], we define a graph G i by where Take an edge e = uv of H. Since f (v 1 ) = 0, Q v1 has at least two vertices.Then Q v1 has a top-two {z 1 , z 2 }.Since G 0 is a line graph, by the above argument on the case of f = 0, there exists an acyclic digraph Since Q v1 has no edges or Q v1 is connected, it follows from Propositions 2.3 and 2.4 that there exists an acyclic digraph D 1 such that C(D 1 ) = G 1 ∪I 2 and D 1 has an acyclic ordering whose first From repeating the process, we can obtain an acyclic digraph Proof of Theorem 1.2:It immediately follows from Theorem 2.5 that the competition number of a generalized line graph is at most two.✷

Generalized line graphs with competition number one
In the following, we show some necessary conditions and sufficient conditions for the competition number of a connected generalized line graph being one.Theorem 1.1 says that the competition number of a connected line graph is one if and only if it has a simplicial vertex.Since the case of the generalized line graph L(H, f ) of a vertex-weighted graph (H, f ) with a zero function f is reduced to Theorem 1.1, we consider the case where f is a nonzero function.In this subsection, when we consider the generalized line graph L(H, f ) of a vertex-weighted graph (H, f ), we denote the cocktail party graph CP (f (v)) added to L(H) by Q v for each vertex v of H in cases where this notation will not cause confusion.
First, we give necessary conditions for the competition number of a connected generalized line graph being one.Proof: Assume that k(G) = 1.Suppose that (i) does not hold.Then f (u) = 1 for any vertex u ∈ V (H).As k(G) = 1 and G is connected, G has a simplicial vertex x by Lemma 2.6.Since  3).Then, the generalized line graph of (H, f ) satisfies both (i) and (ii) of Theorem 2.7.But, the competition number of L(H, f ) is two.
To see this, we recall a result by Kim [3] which states that the deletion of some pendant vertices from a connected graph does not change its competition number if the resulting graph has at least two vertices.Let G ′ be the graph obtained from L(H, f ) by deleting the two vertices in Q v4 and the vertex v ✷ Next, we show the following result which gives sufficient conditions for generalized line graphs to have competition number one.Theorem 2.10 Let G be the generalized line graph of a connected vertex-weighted graph (H, f ) where H has at least one edge and f is a nonzero function.Then, k(G) = 1 if one of the following holds: Let V (Q v4 ) = {q 1 , q 2 , q 3 , q 4 } and E(Q v4 ) = {q 1 q 2 , q 2 q 3 , q 3 q 4 , q 4 q 1 }.To see k(L(H, f )) = 1, we define a digraph D by A(D) := {(q 1 , q 2 ), (q 4 , q 2 ), (v 1 v 2 , q 2 )} ∪ {(q 1 , q 3 ), (q 2 , q 3 ), (v 1 v 2 , q 3 )} ∪{(q 2 , v 1 v 3 ), (q 3 , v 1 v 3 ), (v 1 v 2 , v 1 v 3 )} ∪ {(q 3 , v 1 v 4 ), (q 4 , v 1 v 4 ), (v where z is a new vertex.Then we can easily check that C(D) = L(H, f )∪{z}, and that D is acyclic since the ordering q 1 , q 4 , v 1 v 2 , q 2 , q 3 , v 1 v 3 , v 1 v 4 , z is an acyclic ordering of D. Therefore k(L(H, f )) ≤ 1, Since L(H, f ) is connected, k(L(H, f )) ≥ 1. Hence k(L(H, f )) = 1.✷

Concluding Remark
In this note, we showed that the competition number of a generalized line graph is at most two, which is an extension of a result on the competition number of a line graph.In addition, we tried to characterize generalized line graphs whose competition numbers are one, and then found necessary conditions and sufficient conditions for the competition number of a generalized line graph being one.However, these conditions are not necessary-and-sufficient conditions, so it still remains open to give a complete characterization of generalized line graphs whose competition numbers are one.

Theorem 1 . 1 (
[8]) For a line graph G, k(G) ≤ 2 and the equality holds if and only if G has no simplicial vertex.

Definition 1 Proposition 2 . 4
For a graph G, a set {u, v} of two distinct vertices u and v of G is called a top-two of G if there exists an acyclic digraph D such that C(D) = G ∪ I k(G) and D has an acyclic ordering whose first and second vertices are u and v. ✷ Note that any graph with at least two vertices always has at least one top-two.If a graph G has no edges, then any pair of two vertices of G is a top-two of G. Let G and H be graphs with at least two vertices such that V (G) ∩ V (H) = ∅ and let K be a clique of G. Suppose that there exists an acyclic digraph D ′ such that C(D ′ ) = G ∪ {u 1 , u 2 } where {u 1 , u 2 } is a top-two of H.If either H has no edges or H has no isolated vertices, then there exists an acyclic digraph D satisfying the following: vertices, and (ii) D has an acyclic ordering whose first |V (G)| + 2 terms induce the digraph D ′ .

Theorem 2 . 5
Since the ordering obtained by attaching an acyclic ordering of D ′′ at the end of an acyclic ordering of D ′ gives an acyclic ordering of D, D is acyclic and satisfies (ii).Since H has no isolated vertices, each vertex in H has an out-neighbor in D ′′ .Therefore, each edge of G ⋉ K H between a vertex in K and a vertex of H is an edge of C(D) which results from A * .Thus C(D) = (G ⋉ K H) ∪ I k and so D satisfies (i).✷ For any vertex v of a graph H, let K H (v) denote the set of the edges incident to v in H.Note that K H (v) forms a clique of the line graph of H for each vertex v in a graph H and {K H (v) | v ∈ V (H)} is an edge clique cover of the line graph of H. Let (H, f ) be a vertex-weighted graph.For any edge e = uv of H, there exists an acyclic digraph D such that C(D) = L(H, f ) ∪ I 2 and that N

Lemma 2 . 6 Theorem 2 . 7
If a graph G has competition number one, then G has a simplicial vertex or an isolated vertex.Proof: If G has no simplicial vertex and no isolated vertex, then the competition number of G is at least two by Proposition 2.1.Thus the lemma holds.✷Let G be the generalized line graph of a connected vertex-weighted graph (H, f ) with a nonzero function f .If k(G) = 1, then at least one of the following holds: