Characterization of extremal graphs for certain bounds on 2-independence number

Abstract

A 2-independent set of a graph G = (V;E) is a subset of vertices S V such that the subgraph induced by S has maximum degree less than 2. The 2-independence number of G, denoted by 2(G) is the cardinality of the largest 2-independent set of G. In this thesis, we studied 2-independent sets and the 2-independence number in some particular graphs. Our work aims at the beguining to caracterize graphs with large 2-independent number, we are interested in graphs G satisfying 2(G) = n 􀀀 k with k 2 f1; 2g, then we move to characterize upper bounds for 2, moreover we characterize a lower bound for 2 in trees.