On Editing Graphs into 2-Club Clusters

In this paper, we introduce and study three graph modification problems: 2-Club Cluster Vertex Deletion, 2-Club Cluster Edge Deletion, and 2-Club Cluster Editing. In 2-Club Cluster Vertex Deletion (2-Club Cluster Edge Deletion, and 2-Club Cluster Editing), one is given an undirected graph G and an integer k ≥0, and needs to decide whether it is possible to transform G into a 2-club cluster graph by deleting at most k vertices (by deleting at most k edges, and by deleting and adding totally at most k edges). Here, a 2-club cluster graph is a graph in which every connected component is of diameter 2. We first prove that all these three problems are NP-complete. Then, we present for 2-Club Cluster Vertex Deletion a fixed parameter algorithm with running time O ∗(3.31k ), and for 2-Club Cluster Edge Deletion a fixed parameter algorithm with running time O ∗(2.74k ).

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