Kernels for Edge Dominating Set: Simpler or Smaller

Abstract

A kernelization for a parameterized computational problem is a polynomial-time procedure that transforms every instance of the problem into an equivalent instance (the so-called kernel) whose size is bounded by a function of the value of the chosen parameter. We present new kernelizations for the NP-complete Edge Dominating Set problem which asks, given an undirected graph G = (V,E) and an integer k, whether there exists a subset D ⊆ E with |D| ≤ k such that every edge in E shares at least one endpoint with some edge in D. The best previous kernelization for Edge Dominating Set, due to Xiao, Kloks and Poon, yields a kernel with at most 2 k ² + 2 k vertices in linear time. We first describe a very simple linear-time kernelization whose output has at most 4 k ² + 4 k vertices and is either a trivial “no” instance or a vertex-induced subgraph of the input graph in which every edge dominating set of size ≤ k is also an edge dominating set of the input graph. We then show that a refinement of the algorithm of Xiao, Kloks and Poon and a different analysis can lower the bound on the number of vertices in the kernel by a factor of about 4, namely to $\max\{\frac{1}{2}k^2+\frac{7}{2}k,6 k\}$.