Graphs with equal Independence and Annihilation Numbers

The annihilation number a of a graph is an upper bound of the independence number α of a graph. In this article we characterize graphs with equal independence and annihilation numbers. In particular, we show that α = a if, and only if, either (1) a ≥ n2 and α ′ = a, or (2) a < n2 and there is a vertex v ∈ V (G) such that α(G − v) = a(G), where α is the critical independence number of the graph. Furthermore, we show that it can be determined in polynomial time whether α = a. Finally we show that a graph where α = a is either König-Egerváry or almost König-Egerváry.