The Characterization Of 3-(Total) Domination Critical Graphs Which Are Not Bicritical And Study On P_k-Isolation Number Of Trees

Let G=(V,E)be a graph with the vertex set V and the edge set E.A subset S of vertices in a graph G is a dominating set(total dominating set)of G if every vertex of V\S(V)is adjacent to a vertex in S.The minimum cardinality of a dominating set(total dominating set)is the dominating number(total dominating number),represented as ?(G)(?t(G))of G.G is a 3-?-critical(3-?t-critical)graph if ?(G)=3(?t(G)=3)and ?(G+e)<3(?t(G+e)<3)for any e(?)E.Graph G is bicritical if G-u-v contains a perfect matching for every pair of distinct vertices u,v?V.A graph G is diameter 2-critical if the graph has diameter 2 and the deletion of any edge increases its diameter.Murty and Simon made the conjecture that for a diameter 2-critical graph of order n,size m is at most[n2/4],with equality if and only if G is the complete bipartite graph K[n/2],[n/2],and it is called Murty-Simon conjecture.For any positive k,denote the subset S of vertices in a graph G is a Pk-isolating set of G if G-N[S]contains no Pk,where Pk is a path with length k-1,and the Pk-isolation number ?Pk(G)of G is the size of a smallest set S.In this paper,we first characterize 3-connected 3-?-critical graphs G of even order which are not bicritical:two classes of graphs,which generalizes the result:if the minimum degree of G is at least 4,then G is bicritical[N.Ananchuen and M.D.Plummer,Discrete Math.277(2014)1-13].Next,we characterize 3-?t-critical graphs of even order which are not bicritical.By analyzing the two parameters of diameter and the number of components of G-S,where S is a cutset of G,we get seven classes of graphs.And,we show that Murty-Simon Conjecture holds for the diameter 2-critical graphs of enen order whose complements are not bicritical.Last,we prove that for any n-vertex tree T,if T(?)Pk.then ?pk(G)?n/k+1,and we characterize the n-vertex tree T such that ?Pk(G)=n/k+1.