A set $S\subseteq V(G)$ of a graph $G$ is a dominating set if each vertex has a neighbor in $S$ or belongs to $S$. Let $\gamma(G)$ be the cardinality of a minimum dominating set in $G$. The bondage number $b(G)$ of a graph $G$ is the smallest number of edges $A\subseteq E(G)$, such that $\gamma(G-A)=\gamma(G)+1$. The problem of finding $b(G)$ for a graph $G$ is known to be NP-hard even for bipartite graphs. In this paper, we show that deciding if $b(G)=1$ is NP-hard, while deciding if $b(G)=2$ is coNP-hard, even when $G$ is restricted to one of the following classes: planar $3$-regular graphs, planar claw-free graphs with maximum degree $3$, planar bipartite graphs of maximum degree $3$ with girth $k$, for any fixed $k\geq 3$.

中文翻译：

---

平面图中束缚问题的复杂性

如果每个顶点在 $S$ 中都有一个邻居或属于 $S$，则图 $G$ 的集合 $S\subseteq V(G)$ 是支配集。令 $\gamma(G)$ 是 $G$ 中最小支配集的基数。图$G$的束缚数$b(G)$是最小边数$A\subseteq E(G)$，使得$\gamma(GA)=\gamma(G)+1$。即使对于二部图，为图 $G$ 找到 $b(G)$ 的问题也是 NP-hard 已知的。在本文中，我们表明确定 $b(G)=1$ 是否为 NP-hard，而确定 $b(G)=2$ 是否为 coNP-hard，即使 $G$ 仅限于以下之一类：平面 $3$-正则图，最大度数为 $3$ 的平面无爪图，最大度数为 $3$ 且周长 $k$ 的平面二分图，对于任何固定的 $k\geq 3$。