Research On The Independent Partial Dominating Set Of Graph

Let G=(V,E)be a graph,where V and E are the set of vertices and edges of G respectively.Let S be a subset of vertex set V.If for any v?V\S,V is adjacent to a vertex in S,S is called a dominating set of G.The cardinality of the minimum dominating set of G is called the domination number of G,denoted as ?(G).For a dominating set S of G,if G[S]has no edge,S is said to be an independent dominating set of G.The cardinality of the minimum independent dominating set of graph G is called the independent domination number of G,denoted as i(G).Let F was a graph cluster,a vertex subset S(?)V called F-isolated if H is not a subgraph of G-NG[S]for any H?F.Based on the concept of graph with independent dominating set and F-isolated,the concept of independent partial domination is given.For a vertex subset S of a graph G,if G-NG[S]and S are both independent sets,S is said to be an independent partial dominating set of G.The cardinality of the minimum independent partial dominating set of G is called the independent partial domination number of G,denoted as ip(G),and the independent partial dominating set of G with any cardinality equal to ip(G)is called the ip-set of G.Obviously,the independent dominating set of any G is also an independent partial dominating set of G,namely ip(G)?i(G).In this paper,we first give some properties of graphs with a unique minimum independent partial dominating set,i.e.the effect of deleting vertices or the closed domain of vertices on the uniqueness of ip-set and the size of ip(G)of graphs.Then several operations are given to make two graphs with unique minimum independent partial dominating set that satisfy the condition form a graph with the property of unique minimum independent partial dominating set.Then the uniqueness of ip-set and the specific value of ip(G)are considered on some special graphs.Finally,these properties and operations are used to characterize a tree with a unique minimum independent partial dominating set.