Some bounds on the size of maximum G-free sets in graphs

Abstract

For a given graph [Formula: see text], the independence number [Formula: see text] of [Formula: see text] is the size of the maximum independent set of [Formula: see text]. Finding the maximum independent set in a graph is NP-hard. Another version of the independence number is defined as the size of the maximum-induced forest of [Formula: see text], and called the forest number of [Formula: see text], and denoted by [Formula: see text]. Finding [Formula: see text] is also an NP-hard problem. Suppose that [Formula: see text] is a graph, and [Formula: see text] is a family of graphs, a graph [Formula: see text] has a [Formula: see text]-free [Formula: see text]-coloring if there exists a decomposition of [Formula: see text] into sets [Formula: see text], [Formula: see text], so that [Formula: see text] for each [Formula: see text], and each [Formula: see text]. [Formula: see text] is [Formula: see text]-free, if the subgraph of [Formula: see text] induced by [Formula: see text] is [Formula: see text]-free, i.e., it contains no copy of [Formula: see text]. Finding a maximum subset [Formula: see text] of [Formula: see text], such that [Formula: see text] is a [Formula: see text]-free graph is a very hard problem as well. In this paper, we study the generalized version of the independence number of a graph. Also, we give some bounds about the size of the maximum [Formula: see text]-free subset of graphs as another purpose of this paper.