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 $k\geq 3$.
翻译:$S\ subseteq V( G) 美元 。 如果每个顶点有一个以美元为单位的相邻方, 或属于美元为美元。 请将$gamma( G) 美元作为以美元为单位的最低支配量的基数。 图形G$的质役号为$b( G) 美元, 是一个以美元为单位的最小边缘数 $A\ subseteq E( G) 美元, 诸如$gamma( G- A) ⁇ gamma( G)+1 美元。 找到美元为美元为单位的相邻方$b( G) 或属于美元。 即使当美元被限制在以下类别之一时, 也很难找到美元: 平面为3美元、 平面为3美元、 平面为3美元、 平面为3美元、 固定为3美元。