A polynomial kernel for Block Graph Deletion

TL;DR

This paper presents a polynomial kernel of size O(k^6) for the Block Graph Deletion problem, enabling more efficient preprocessing, and introduces the concept of 'complete degree' for kernelization analysis.

Contribution

It provides the first polynomial kernel for Block Graph Deletion and introduces the 'complete degree' notion, advancing kernelization techniques for graph modification problems.

Findings

Kernel with O(k^6) vertices for Block Graph Deletion

Polynomial-time algorithm with 10^k * n^{O(1)} complexity

Implication for polynomial kernels in related graph classes

Abstract

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