Competition Numbers of a Kind of Pseudo-Halin Graphs

For any graph G , G together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number ( ) k G of a graph G is defined to be the smallest number of such isolated vertices. In general, it is hard to compute the competition number ( ) k G for a graph G and characterizing a graph by its competition number has been one of important research problems in the study of competition graphs. A 2-connected planar graph G with minimum degree at least 3 is a pseudo-Halin graph if deleting the edges on the boundary of a single face 0 f yields a tree. It is a Halin graph if the vertices of 0 f all have degree 3 in G . In this paper, we compute the competition numbers of a kind of pseudo-Halin graphs.


Introduction and Preliminary
Let ( ) , G V E = be a graph in which V is the vertex set and E the edge set.We always use V and E to denote the vertex number and the edge number of G , respectively.The notion of competition graph was introduced by Cohen [1] in connection with a problem in ecology.Let uv E G ∈ if and only if there exists some vertex ( ) k G of a graph G is defined to be the smallest number k such that G together with k isolated vertices added is the competition graph of an acyclic digraph.It is difficult to compute the competition number of a graph in general as Opsut [3] has shown that the computation of the competition number of a graph is an NP-hard problem.But it has been one of important research problems in the study of competition graphs to characterize a graph by its competition number.Recently, many papers related to graphs' competition numbers appear.Kim et al. [4] studied the competition numbers of connected graphs with exactly one or two triangles.Sano [5] studied the competition numbers of regular polyhedra.Kim et al. [6] studied the competition numbers of Johnson graphs.Park and Sano [7] [8] studied the competition numbers of some kind of hamming graphs.Kim et al. [9] studied the competition numbers of the complement of a cycle.Furthermore, there are some papers (see [10] [11] [12] [13] [14]) focused on the competition numbers of the complete multipartite graphs, and some papers (see [15]- [21]) concentrated on the relationship between the competition number and the number of holes of a graph.A cycle of length at least 4 of a graph as an induced subgraph is called a hole of the graph.We use r I to denote the graph consisting only of r isolated vertices, and r G I ∪ the disjoint union of G and r I .

[ ]
G S of G is a subgraph of G whose vertex set is S and whose edge set is the set of those edges of G that have both ends in S .
All graphs considered in this paper are simple and connected.For a vertex v in a graph G, let the open neighborhood of v be denoted by For any set S of vertices in G , we define the neighborhood of S in G to be the set of all vertices adjacent to vertices in S , this set is denoted by ( ) ( )

Lemmas
We first introduce two useful Lemmas.
Lemma 1 (Harary et al. [22]).Let ( ) be a digraph.Then D is acyclic if and only if there exists an ordering of vertices, [ ] , such that one of the following two conditions holds: By this lemma, if D is an acyclic digraph, we can find a vertex labelling . We call σ an acyclic labelling of D .Conversely, if D is a digraph with an acyclic labelling, then D is acyclic.
Lemma 2 (Kim and Roberts [4]).For a tree T and a vertex v of T , there is an acyclic digraph D so that is the competition graph of D for 0 v not in T and so that v has only outgoing arcs in D .
Kim and Roberts [4] proved Lemma 2 by the following algorithm.
, and ( ) for some vertex 0 v not in T , and ( ) ( In the procedure, we may avoid selecting v until we select all other vertices since there are at least two vertices of degree 1 in a tree with more than one vertex.
In fact, this algorithm provides an acyclic labelling ( ) where ( ) Opsut [3] gave the following two lower bounds for the competition number of a graph.
Theorem 1 (Opsut [3]).For any graph G , ( ) ( ) ( ) Theorem 2 (Opsut [3]).For any graph G , ( ) ( ) ( ) Lemma 3.For any generalized Halin graph be a generalized Halin graph, where T and C are the characteristic tree and the adjoint cycle of G , respectively.Suppose that along cycle C by clockwise order we can partition ( ) and the vertices in each ( ) V C are con- secutive on C , where 1 i k ≤ ≤ .Let i u be the common neighbor of the vertices in ( ) , where 1 i k ≤ ≤ .We assume that the vertices in ( ) , where and We assume that the vertices in ( ) , , , , where , and if ( ) V C ≠ ∅ then we always let ( ) and arbitrarily select a vertex in ( ) By Lemma 1 and the algorithm in the proof of Lemma 2, we can construct an acyclic is the competition graph of D for 0 v not in T , and get an acyclic labelling , where 1 and 1 ; , where 1 and 1 .
, and we always have ( ) and whose arc set is ( ) D be a digraph whose vertex set is and whose arc set is x x β + = , and all 1 j y ( ) are new added vertices.Case 3.
( ) , , , and whose arc set is are new added vertices.We note that 1 D , 2 D and 3 D are acyclic.This is because every arc added here goes either from a big label vertex to a small label vertex or from a vertex in ( ) , , , and , 0 , , , .
And we know that be a not 4 K generalized Halin graph, where T and C are the characteristic tree and the adjoint cycle of G , respectively.Since ( ) ( )  just includes all triangles in G and the edges in C , so ( )  has not any common edges.So we have

Pseudo-Halin Graph
Now we consider a pseudo-Halin graph G with the exterior face 0 f and ( ) { } . Let u and v be the neighbors of x on the boundary of 0 f .Without lose of generality, we may always assume that u is on the left of x and v on the right of ∪ , where T and C′ are the characteristic tree and the adjoint cycle of G′ , respectively.Then it is easy to see that C′ is got from the boundary of 0 f by deleting the edges xu and xv , and adding the edge uv .So we have ( ) ( ) . Let b x ≠ be another neighbor of u on cycle C′ .The characteristic tree T of G′ is just the tree got from G by deleting the edges on the boundary of the face 0 f .So T may also be called the characteristic tree of G .Let u′ be the neighbor of u in T and v′ the of v in T , respectively.
We construct a graph G′′ from G′ by replacing the edge uv by a path ux v ′ , and joining x with x′ .It is not difficult to see that G′′ is a Halin graph.Since every Halin graph contains a triangle, and x′ is not a vertex of any triangle in G′′ , then G′ also contains a triangle.Therefore Proof.Suppose that G is a pseudo-Halin graph with ( ) { } ( ) where C′ is the adjoint cycle of graph v v = and by the similar way used in the proof of Lemma 3, there is a digraph D′ such that where 0 v and 1 v are two isolated vertices not in G′ .Note that See Figure 2. Note that D is acyclic since every arc added here goes from a big label vertex to a small label vertex.It is easy to see that ( ) On the other hand, since for each vertex note that the maximal clique in G is a triangle, so we have , and C′ the adjoint cycle of graph Proof.Suppose that G is a pseudo-Halin graph with ( ) { } ( ) where C′ is the adjoint cycle of graph Denote and labelling the vertices of G′ in a similar way as used in Lemma 3. By Lemma 3, ( ) ( ) + , and there is a digraph D′ such that ( ) ( ) V C′ + isolated vertices not in G′ .By the similar way used in the proof of Lemma 3, there exists a vertex where w is a new added vertex to D′ .Note that D is acyclic since every arc added here goes from a big label vertex to a small label vertex or to a new added vertex.It is easy to see that ( ) ( ) ) , , where Proof. 1) ( ) Obviously, { } , x u and { } , 2) ( ) It is easy to see that { } , , x u u′ and { } , , x v v′ are two maximal cliques of G .Note that uv is a maximal clique of G′ . 3) Without lose of generality, we just need to consider the case xu E ′ ∈ but xv E ′ ∉ .
By the proof of Case ( 1 .Note that G′ can not be 4 K , so by Lemmas 4 we have On the other hand, by Theorem 1 we have So by lemmas 6, we have the following result.

Concluding Remarks
In this paper, we study the competition numbers of a kind of pseudo-Halin graphs with just one irregular vertex.

C D C D
= for a given digraph D , and we note that multitype graphs can be used to study the multi-species in ecology and have been deeply studied, see [23] [24].So these generalizations of competition graphs may be more realistic and more interesting.
in which V is the vertex set and A the set of directed arcs.The competition graph ( ) C D of D is the undirected graph G with the same vertex set as D and with an edge ( )

1 V 2 V
of a vertex v in digraph D is a vertex u such that ( ) ( ) , u v A D ∈ ; an out-neighbor of a vertex v is a vertex w such that ( ) ( ) , v w A D ∈ .We denote the sets of in-neighbors and out-neighbors of v in D by S of the vertex set of a graph G is called a clique of G if [ ] G S is a complete graph.For a clique S of a graph G and an edge e of G , we say e is covered by S if both of the endpoints of e are contained in S .An edge clique cover of a graph G is a family of cliques such that each edge of G is covered by some clique in the family.The edge clique cover number ( ) e G θ 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 such that each vertex of G is contained in some clique in the family.The vertex clique cover number ( ) v G θ of a graph G is the minimum size of a vertex clique cover of G .A generalized Halin graph G T C = ∪ is a plane graph consisting of a plane embedding of a tree T and a cycle C connecting the leaves (vertices of degree 1) of T such that C is the boundary of the exterior face.The tree T and the cycle C are called the characteristic tree and the adjoint cycle of G , respectively.For each ( ) ∈ , then we called v a simple leaf of T , otherwise, a compound leaf of T .Denote all simple leaves of T by ( ) C and all compound leaves of T by ( ) C , respectively.It is obvious that

and 0 1 ,
v v are new added vertices.Case 2.

( ) 2 V
C′ ≠ ∅ .So we just need to consider the following cases.Theorem 3. Let G be a pseudo-Halin graph with ( ) { } and labelling the vertices of G′ in a similar way as used in Lemma 3. By Lemma 3, ( ) 2 k G′ ≤ .Let1 1

 Lemma 6 .
Let G be a pseudo-Halin graph with ( ) { } If are all maximal cliques of graph G′ .So are two maximal cliques of graph G′ , but { } , are all maximal cliques of graph G′ , but { } are all maximal cliques of graph G′ .If ( )2 v V C′ ∈ , then { } , x v′ is a maximal clique of graph G′ , but { } , v v′ not.So the result follows.For a pseudo-Halin graph G , suppose that 0 f be the exterior face of G and

Theorem 4 . 1 V
Let G be a pseudo-Halin graph with ( ) { } C′ ≠ ∅ , where C′ is the adjoint cycle of graph {

2 k
For a pseudo-Halin graph G with the exterior face 0 f and ( )0 1 I f = , we showthat if all leaves of the characteristic tree of G are compound leaves, then ( ) generalized Halin graph G is a Halin graph when each interior vertex of G has degree at least 3.A 2-connected planar graph G with minimum degree at least 3 is a pseudo-Halin graph if deleting the edges on the boundary of a single face 0 f yields a tree.It is a for some cases, but we can not provide the accurate value of the competition number of G for other cases.So it would be valuable to get the accurate value of the competition number of the pseudo-Halin graph with just one irregular vertex, and it may be interesting to study competition numbers of general pseudo-Halin graphs.