Admissible Property of Graphs in Terms of Radius

Abstract

Let G be a graph and ${\mathcal {P}}$ be a property of graphs. A subset $S \subseteq V(G)$ is called a ${\mathcal {P}}$-admissible set of G if $G-N[S]$ admits the property ${\mathcal {P}}$. The ${\mathcal {P}}$-admission number of G, denoted by $\eta (G,{\mathcal {P}})$, is the cardinality of a minimum ${\mathcal {P}}$-admissible set in G. For a positive integer k, we say a graph G has the property ${\mathcal {R}}_k$ if the radius of each component of G is at most k. In particular, $\eta (G,{\mathcal {R}}_1)$ 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 $\eta (G,{\mathcal {R}}_1)$ for a connected graph G. We show that for a connected graph $G\ne C_7$ of order n, $\eta (G,{\mathcal {R}}_1)\le \frac{n}{4}$. The bound is sharp. Several related problems are proposed.