The Graphs Determined By an Adjacency Property

We determine all graphs \(G\) of order at least \(k + 1\), \(k \geq 3\), with the property that for any \(k\)-subset \(S\) of \(V(G)\) there is a unique vertex \(x, x \in V(G) – S\), which has exactly two neighbours in \(S\). Such graphs have exactly \(k + 1\) vertices and consist of a family of vertex-disjoint cycles. When \(k = 2\) it is clear that graphs with this property are the so-called friendship graphs.