Characterization Of Claw-free 3-regular Graphs With The Domination Number Equaling To The Edge Domination Number

Let G=（V,E）be a graph with the vertex set V and the edge set E.Set S?V is a dominating set satisfying that each vertex not in S has a neighbor in S.The domination number of G is the minimum cardinality of dominating set.The edge dominating set S’ is a set of edges of graph G such that each edge not in S’ has a neighbor in S’.The edge domination number of a graph is the minimum cardinality of edge dominating set,denoted by γ’（G）.The degree of a vertex v∈V,denoted by d（v）,is the number of edges incident with vertex v.A graph G is k-regular if d（v）=k for all v∈V.Let Kn denote the complete graph of n vertex.A complete graph K3 is also called a triangle.The complete graph on four vertices minus one edge is called a diamond.A graph G is F-free if it does not contain F as an induced subgraph.In particular,if F=K1,3,then we say that the graph is claw-free.Henning etl.[2]pointed out that:If G is a claw-free 3-regular graph,then V can be divided to some subsets,which can induces a triangle or a diamond,and we call this partition a △-D-partiton.Suppose the number of triangle and diamond induced by △-D-partiton is t and d,respectively.In this paper we study the claw-free 3-regular graph G.Firstly,we propose the concept of a pseudo-path,considering the △-D-partiton of G,study the pseudo-path of triangles induced by △-D-partiton,get γ（G）≤t+d and basing on this prove the necessary and sufficient condition of satisfying γ（G）=t+d.Then,in general,γ（G）≤γ’（G）,under the properties domination number and edge domination number,by modifying the minimum dominating set and minimum edge dominating set,proved that γ（G）<γ’（G）,get the necessary condition of satisfying γ（G）=γ’（G）.Lastly,this paper give the graph class F and the components of R with the relationships between these components,by discussing different relationships,characterize the claw-free 3-regular graphs which satisfyingγ（G）=γ’（G）.