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The $2$-domination number $\gamma_2(G)$ of a graph $G$ is the minimumcardinality of a set $S\subseteq V(G)$ such that every vertex from$V(G)\setminus S$ is adjacent to at least two vertices in $S$. The annihilationnumber $a(G)$ 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 itsedges. It was conjectured that $\gamma_2(G) \leq a(G) +1$ holds for everyconnected graph $G$. The conjecture was earlier confirmed, in particular, forgraphs 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 bearbitrarily larger than the annihilation number. On the positive side we provethe conjectured bound for a large subclass of bipartite, connected cacti, thusgeneralizing a result of Jakovac from [Discrete Appl.\ Math.\ 260 (2019)178--187].

The annihilation number does not bound the 2-domination number from the above