  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 $\mathcal{O}(k^{6})$ 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 $10^{k}\cdot
n^{\mathcal{O}(1)}$.
