Admissible Property of Graphs in Terms of Radius

Let G be a graph and P be a property of graphs. A subset S ⊆ V ( G ) is called a P - admissible set of G if G - N [ S ] admits the property P . The P - admission number of G , denoted by η ( G , P ) , is the cardinality of a minimum P -admissible set in G . For a positive integer k , we say a graph G has the property R k if the radius of each component of G is at most k . In particular, η ( G , R 1 ) is the cardinality of a smallest set S such that each component of G - N [ S ] has a universal vertex. In this paper, we establish sharp upper bound for η ( G , R 1 ) for a connected graph G . We show that for a connected graph G ≠ C 7 of order n , η ( G , R 1 ) ≤ n 4 . The bound is sharp. Several related problems are proposed.