On an annihilation number conjecture^∗

Let α(G) denote the cardinality of a maximum independent set, while µ(G) be the size of a maximum matching in the graph G = (V, E). If α(G) + µ(G) = |V |, then G is a König-Egerváry graph. If d₁ ≤ d₂ ≤ · · · ≤ d_n is the degree sequence of G, then the annihilation number a (G) of G is the largest integer k such that ^Pk_i=1 d_i ≤ |E|. A set A ⊆ V satisfying ^P_v∈_A deg(v) ≤ |E| is an annihilation set; if, in addition, deg (x) + ^P_v∈_A deg(v) > |E|, for every vertex x ∈ V (G) − A, then A is a maximal annihilation set in G. In 2011, Larson and Pepper conjectured that the following assertions are equivalent: (i) α (G) = a (G); (ii) G is a König-Egerváry graph and every maximum independent set is a maximal annihilating set. It turns out that the implication “(i) =≻ (ii)” is correct. In this paper, we show that the opposite direction is not valid, by providing a series of generic counterexamples.