The 2-domination number γ 2 (G) of a graph G is the minimum cardinality of a set S ⊆ V (G) such that every vertex from V (G) \ S is adjacent to at least two vertices in S. The annihilation number a(G) is the largest integer k such that the sum of the first k terms of the non-decreasing degree sequence of G is at most the number of its edges. It was conjectured that γ 2 (G) ≤ a(G) + 1 holds for every connected graph G. The conjecture was earlier confirmed, in particular, for graphs of minimum degree 3, for trees, and for block graphs. In this paper, we disprove the conjecture by proving that the 2-domination number can be arbitrarily larger than the annihilation number. On the positive side we prove the conjectured bound for a large subclass of bipartite, connected cacti, thus generalizing a result of

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