Partitioning a graph into alliance free sets

A strong defensive alliance in a graph G = ( V , E ) is a set of vertices A ⊆ V , for which every vertex v ∈ A has at least as many neighbors in A as in V − A . We call a partition A , B of vertices to be an alliance-free partition, if neither A nor B contains a strong defensive alliance as a subset. We prove that a connected graph G has an alliance-free partition exactly when G has a block that is other than an odd clique or an odd cycle.