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

The $2$-domination number $\gamma_2(G)$ of a graph $G$ is the minimum cardinality 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 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 $\gamma_2(G) \leq a(G) +1$ 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 Jakovac from [Discrete Appl.\ Math.\ 260 (2019) 178--187].