Research On Vertex Edge Domination And Total Vertex Edge Domination Of Graph

Let G=(V,E)be a graph where V is the set of vertices of G and E is the set of edges of G.Let S be a subset of V.The subset S is a dominating set of G if every vertex of V\S is adjacent to a vertex in S.A subset S(?)V is a total dominating set of G if S is a dominating set of G and the subgraph induced by S has no isolated vertices,that is,every vertex of S has a neighbour in S.The minimum cardinality of a dominating set(total dominating set)is the domination number(total domination number),denoted by ?(G)(?t(G)).A vertex u ? V vertex-edge dominates an edge vw?E if 1.u=v or u=w(u is incident to vw),or 2.uv or uw is an edge in G(u is incident to an edge that is adjacent to vw).In other words,a vertex u vertex-edge dominates the edges incident to vertices in N[u].A set S(?)V is a vertex-edge dominating set if for all edges e ?E,there exists a vertex v ? S that vertex-edge dominates e.A subset S(?)V is a total vertex-edge dominating set of G if S is a vertex-edge dominating set of G and the subgraph induced by S has no isolated vertices,that is,every vertex of S has a neighbour in S.The minimum cardinality of a vertex-edge dominating set(total vertex-edge dominating set)is the vertex-edge domination number(total vertex-edge domination number),denoted by ?ve(G)(?vet(G)).In this paper,we first give the algorithm to find the vertex-edge domination number of tree and the total vertex-edge dominating set of tree.Then we characterize the tree satisfying the vertex-edge domination number is equal to domination number:Tree T that satisfies ?ve(T)=?(T)if and only if T is K1,n,n?1 or T is r K1,n?2 are connected by leaf nodes and satisfy that each K1,n has at least one leaf node that is not connected to other leaf nodes.Then,we characterize the circle satisfies the condition that the total vertex-edge domination number is equal to double vertex-edge domination number:the circle Cn satisfies ?ve(Cn)=2?ve(Cn)if and only if n ? {2,3,4,7,8,12}.Finally,we give the characterization of the tree whose total vertex-edge domination number is equal to total vertex-edge domination number:by constructing the tree class H and L,we study the relationship between them and draw a series of conclusions.Finally,we get the tree class whose total vertex-edge domination number is equal to total vertex-edge domination number.