Computing the independence number of dense triangle-free graphs

Abstract

Computing the independence number of a graph remains NP-hard, even restricted to the class of triangle-free graphs. So the question arises, whether this remains valid if the minimum degree is required to be large. While in general graphs this problem remains NP-hard even within the class of graphs with minimum degree δ > (1 − e)n, the situation is different for triangle-free graphs. It will be shown that for triangle-free graphs with δ > n/3 the independence number can be computed as fast as matrix multiplication, while within the class of triangle-free graphs with δ > (1-e)n/4 the problem is already NP-hard.