A polynomial kernel for Block Graph Vertex Deletion

In the Block Graph Deletion problem, we are given a graph G on n vertices and a positive integer k, and the objective is to check whether it is possible to delete at most k vertices from G to make it a block graph, i.e., a graph in which each block is a clique. In this paper, we obtain a kernel with Opkq vertices for the Block Graph Deletion problem. This is a first step to investigate polynomial kernels for deletion problems into nontrivial classes of graphs of bounded rank-width, but unbounded tree-width. Our result also implies that Chordal Vertex Deletion admits a polynomial-size kernel on diamond-free graphs. For the kernelization and its analysis, we introduce the notion of ‘complete degree’ of a vertex and also use the Sauer-Shelah Lemma [J. of Comb. Theory, Ser. A, 13:145–147, 1972]. We believe that these are of independent interest and the underlying ideas can be potentially applied to other problems. We also prove that the Block Graph Deletion problem can be solved in time 10 ̈ n.