A 9k Kernel for Nonseparating Independent Set in Planar Graphs

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 − ǫ)k vertices, for any ǫ > 0, assuming P = NP. We also show a very simple 5k-vertices kernel for Planar Max Leaf, which results in a lower bound of (5/4 − ǫ)k vertices for the kernel of Planar Connected Dominating Set (also under P = 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 9 n). The result (a) is a generalization of a theorem of Kleitman and West [8] 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|.

Extension and its price for the connected vertex cover problem

“…They also propose a 5 3 -approximation algorithm for Min CVC for any class of graphs where Min VC is polynomial-time solvable. Parameterized complexity for Min CVC and Max NSIS have been studied in [23,24] while the enumeration of minimal connected vertex covers is investigated in [18] where it is shown that the number of minimal connected vertex covers of a graph of n vertices is at most 1.8668 n , and these sets can be enumerated in time O(1.8668 n ). For chordal graphs (even for chordality at most 5), the authors are able to give a better upper bound.…”

Extension and Its Price for the Connected Vertex Cover Problem

“…They also propose a 5 3 -approximation algorithm for Min CVC for any class of graphs where Min VC is polynomial-time solvable. Parameterized complexity for Min CVC and Max NSIS have been studied in [23,24] while the enumeration of minimal connected vertex covers is investigated in [18] where it is shown that the number of minimal connected vertex covers of a graph of n vertices is at most 1.8668 n , and these sets can be enumerated in time O(1.8668 n ). For chordal graphs (even for chordality at most 5), the authors are able to give a better upper bound.…”