Maximum independent sets near the upper bound

Abstract

The size of a largest independent set of vertices in a given graph G is denoted by α(G) and is called its independence number (or stability number). Given a graph G and an integer K, it is NP-complete to decide whether α(G)≥K. An upper bound for the independence number α(G) of a given graph G with n vertices and m edges is given by α(G)≤p≔⌊12+14+n2−n−2m⌋.In this paper we will consider maximum independent sets near this upper bound. Our main result is the following: There exists an algorithm with time complexity O(n2) that, given as an input a graph G with n vertices, m edges, p≔⌊12+14+n2−n−2m⌋, and an integer k≥0 with p≥2k+1, returns an induced subgraph Gp,k of G with n0≤p+2k+1 vertices such that α(G)≤p−k if and only if α(Gp,k)≤p−k. Furthermore, we will show that we can decide in time O(1.27383k+n2) whether α(Gp,k)≤p−k.