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 $$10^{k}\cdot n^{{\mathcal {O}}(1)}$$ .
Highlights
In parameterized complexity, an instance of a parameterized problem consists in a pair px, kq, where k is a secondary measurement, called the parameter
A parameterized problem Q Ď Σ N is fixed-parameter tractable (FPT ) if there is an algorithm which decides whether px, kq belongs to Q in time f pkq |x|Op1q for some computable function f
A parameterized problem is said to admit a polynomial kernel if there is a polynomial time algorithm in |x| ` k, called a kernelization algorithm, that reduces an input instance into an instance with size bounded by a polynomial function in k, while preserving the Yes/No answer
Summary
An instance of a parameterized problem consists in a pair px, kq, where k is a secondary measurement, called the parameter. Marx [20] firstly showed that the Chordal Vertex Deletion problem is FPT, and Cao and Marx [3] improved that it can be solved in time 2Opk log kqnOp1q It remains open whether there is a single-exponential FPT algorithm or a polynomial kernel [20, 3]. It is known to be FPT from the meta-theorem on graphs of bounded rank-width [6], but for our knowledge, it is open whether there is a single exponential FPT algorithm or a polynomial kernel for this problem even for w “ 1. On diamond-free graphs, Chordal Vertex Deletion admits a kernel with Opk6q vertices and can be solved in time Op10knOp1qq
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