Independent sets near the lower bound in bounded degree graphs

TL;DR

This paper characterizes graphs with independence numbers close to the lower bound given by Brook's Theorem and provides a kernelization result for the independent set problem in such graphs.

Contribution

It offers an approximate structural characterization of graphs near the lower bound of independence number and introduces a kernelization approach for the independent set problem.

Findings

Graphs with independence number close to the lower bound are structurally characterized.

The independent set problem admits a kernel of size O(k) in these graphs.

The results improve understanding of independent sets in bounded degree graphs.

Abstract

By Brook's Theorem, every n-vertex graph of maximum degree at most Delta >= 3 and clique number at most Delta is Delta-colorable, and thus it has an independent set of size at least n/Delta. We give an approximate characterization of graphs with independence number close to this bound, and use it to show that the problem of deciding whether such a graph has an indepdendent set of size at least n/Delta+k has a kernel of size O(k).