A 9k kernel for nonseparating independent set in planar graphs

TL;DR

This paper introduces a kernelization technique reducing the Planar Maximum Nonseparating Independent Set problem to at most 9k vertices, providing new bounds and insights for related planar graph problems and extremal graph theory.

Contribution

It presents a 9k-vertex kernel for the problem, establishes lower bounds for related problems, and proves new extremal graph properties involving nonseparating sets and spanning trees.

Findings

A 9k-vertex kernel for Planar Maximum Nonseparating Independent Set.

Lower bounds of (9/8 - epsilon)k and (5/4 - epsilon)k for related problems assuming P ≠ NP.

New extremal results on spanning trees and independent sets in graphs without degree-two separators.

Abstract

We study kernelization (a kind of efficient preprocessing) for NP-hard problems on planar graphs. Our main result is a kernel of size at most 9k vertices for the Planar Maximum Nonseparating Independent Set problem. A direct consequence of this result is that Planar Connected Vertex Cover has no kernel with at most (9/8 - epsilon)k vertices, for any epsilon > 0, assuming P \ne NP. We also show a very simple 5k-vertices kernel for Planar Max Leaf, which results in a lower bound of (5/4 - epsilon)k vertices for the kernel of Planar Connected Dominating Set (also under P \ne NP). As a by-product we show a few extremal graph theory results which might be of independent interest. We prove that graphs that contain no separator consisting of only degree two vertices contain (a) a spanning tree with at least n/4 leaves and (b) a nonseparating independent set of size at least n/9 (also, equivalently, a connected vertex cover of size at most 8/9n). The result (a) is a generalization of a theorem of Kleitman and West [SIDMA 1991] who showed the same bound for graphs of minimum degree three. Finally we show that every n-vertex outerplanar graph contains an independent set I and a collection of vertex-disjoint cycles C such that 9|I| >= 4n-3|C|.