The 2-admissible property of graphs in terms of radius

Let G be a graph and N_G^2[D] be the 2-closed neighborhood of a vertex subset D in G . For a property P of a graph G , a vertex subset D is said to be a P -2-admissible set of G if G - N_G^2[D] admits the property P . The P -2-admission number of G , denoted by η(G, P, 2) , is the cardinality of a minimum P -2-admissible set in G . For a positive integer k , we say a graph G has the property R_k if each component of G has radius at most k . Then the R_1 -2-admission number of a graph G , denoted by η(G, R_1, 2) , is the cardinality of a minimum vertex subset D such that V(G) = N_G^2[D] or each component of G - N_G^2[D] is a graph having radius at most one. In this paper, we prove that if G is a connected graph of order n such that G ∉ \{P_4, C_4, C_9\} , then η(G, R_1, 2) ≤ n/5 , and that the bound is sharp.