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].

中文翻译：

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湮灭数不限制上面的2-支配数

图 $G$ 的 $2$-支配数 $\gamma_2(G)$ 是集合 $S\subseteq V(G)$ 的最小基数，使得 $V(G)\setminus S$ 中的每个顶点是与 $S$ 中的至少两个顶点相邻。湮灭数$a(G)$是最大的整数$k$，使得$G$的度数非递减序列的前$k$项之和至多是它的边数。推测 $\gamma_2(G) \leq a(G) +1$ 对每个连通图 $G$ 成立。该猜想早先得到了证实，特别是对于最小度为 $3$ 的图、树和块图。在本文中，我们通过证明$2$-支配数可以任意大于湮灭数来反驳这个猜想。从积极的方面来说，我们证明了二分、连通仙人掌的一个大子类的猜想界限，