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 ?'(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 ?(G)?t+d and basing on this prove the necessary and sufficient condition of satisfying ?(G)=t+d.Then,in general,?(G)??'(G),under the properties domination number and edge domination number,by modifying the minimum dominating set and minimum edge dominating set,proved that ?(G)