Abstract :
A function f : V→{−1,1} defined on the vertices of a graph G=(V,E) is a signed 2-independence function if the sum of its function values over any closed neighbourhood is at most one. That is, for every v∈V, f(N[v])⩽1, where N[v] consists of v and every vertex adjacent to v. The weight of a signed 2-independence function is f(V)=∑f(v), over all vertices v∈V. The signed 2-independence number of a graph G, denoted αs2(G), equals the maximum weight of a signed 2-independence function of G. In this paper, we establish upper bounds for αs2(G) in terms of the order and size of the graph, and we characterize the graphs attaining these bounds. For a tree T, upper and lower bounds for αs2(T) are established and the extremal graphs characterized. It is shown that αs2(G) can be arbitrarily large negative even for a cubic graph G.
