A characterization of trees for new lower bound on the k-independence number

Let G be a simple graph with vertex set V and edge set E. The number of vertices of G is called the order, and is denoted by n = n(G). The open neighborhood N(v) = NG(v) of a vertex v consists of all vertices adjacent to v and d(v) = dG(v) = |N(v)| is the degree of v. The closed neighborhood of a vertex v is defined by N [v] = NG [v] = NG (v)∪{v}. By ∆ = ∆(G), we denote the maximum degree of the graph G. A vertex of degree one is called a leaf and its neighbor is called a support vertex. If v is a support vertex then Lv will denote the set of the leaves adjacent to v. If v is support vertex with |Lv| ≥ 2, then v is called strong support, else v is called weak support. We denote by S(G) and L(G) the set of