Nonblocker in H-Minor Free Graphs: Kernelization Meets Discharging

Abstract

Perhaps the best known kernelization result is the kernel of size 335k for the Planar Dominating Set problem by Alber et al. [1], later improved to 67k by Chen et al. [5]. This result means roughly, that the problem of finding the smallest dominating set in a planar graph is easy when the optimal solution is small. On the other hand, it is known that Planar Dominating Set parameterized by k′ = |V| − k (also known as Planar Nonblocker) has a kernel of size 2k′. This means that Planar Dominating Set is easy when the optimal solution is very large. We improve the kernel for Planar Nonblocker to \(\frac{7}{4}k'\). This also implies that Planar Dominating Set has no kernel of size at most \((\frac{7}{3}-\epsilon)k\), for any ε > 0, unless P = NP. This improves the previous lower bound of (2 − ε)k of [5]. Both of these results immediately generalize to H-minor free graphs (without changing the constants).

In our proof of the bound on the kernel size we use a variant of the discharging method (used e.g. in the proof of the four color theorem). We give some arguments that this method is natural in the context of kernelization and we hope it will be applied to get improved kernel size bounds for other problems as well.

As a by-product we show a result which might be of independent interest: every n-vertex graph with no isolated vertices and such that every pair of degree 1 vertices is at distance at least 5 and every pair of degree 2 vertices is at distance at least 2 has a dominating set of size at most \(\frac{3}7n\).