CN-TST 2012 Problem 7

In a simple graph $G$ , we call $t$ pairwise adjacent vertices a $t$ -clique. If a vertex is connected with all other vertices in the graph, we call it a central vertex. Given are two integers $n,k$ such that $\dfrac {3}{2} \leq \dfrac{1}{2} n < k < n$ . Let $G$ be a graph on $n$ vertices such that**(1)** $G$ does not contain a $(k+1)$ -clique;(2) if we add an arbitrary edge to $G$ , that creates a $(k+1)$ -clique. Find the least possible number of central vertices in $G$ .