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 O(k6) vertices for the Block Graph Deletion problem.
This is a first step to investigate polynomial kernels for deletion problems
into non-trivial 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. We believe
that the underlying idea can be potentially applied to other problems. We also
prove that the Block Graph Deletion problem can be solved in time 10k⋅nO(1).Comment: 22 pages, 2 figures, An extended abstract appeared in IPEC201