On the hardness of problems around s-clubs on split graphs
Résumé
We investigate the complexity of problems related to s-clubs. Given a graph, an s-club is a subset of vertices such that the subgraph induced by it has diameter at most s. We show that partitioning a split graph into two 2-clubs is NP-hard. Moreover, we prove that finding the minimum number of edges to add to a split graph in order to obtain a diameter of at most 2 is W[2]-hard with respect to the number of edges to add. Finally we show that finding the minimum number of edges to keep within a split graph of diameter 2 or 3 in order to maintain its diameter is NP-complete.