Characterization For Trees And Unicyclic Graphs With Equal Neighborhood Total Domination Number And Connected Domination Number

Let G =(V,E)be a graph of order n with without isolated vertices.The subset S of V is a dominating set if every vertex in V\S is adjacent to some vertex in S.A dominating set S of G is called a neighborhood total dominating set if G[N(S)]has no isolated vertices.The minimum cardinality of a neighborhood to-tal dominating set of G is called the neighborhood total domination number of G and is denoted by ?nt(G).A neighborhood total dominating set with cardinality of ?nt(G)is called a ?nt(G)-set.A dominating set S of G is called a connected dominating set if G[S]is connected.The connected domination number ?c(G)of G is the minimum cardinality taken over all minimal connected dominating sets of G.In this paper,firstly,we class trees with equal neighborhood total domi-nation number and connected domination number according to |I(T)|.Secondly,according to the classes of trees with equal neighborhood total domination number and connected domination number that we have proved.Then,we class unicyclic graphs with equal neighborhood total domination number and connected dom-ination number according to |X|.Finally,we class unicyclic graphs with equal neighborhood total domination number and connected domination number with condition |X|? 4 according to the longest path t in G[X].