call $N(v)\coloneqq \{x\in V\mid(x,v)\in E\}$ the *open neighborhood* of $v$. Further, $N[v]\coloneqq N(v)\cup\{v\}$ is the *closed neighborhood* of $v$. We also use these terms for sets $S\subseteq V$, where $N(S)\coloneqq \bigcup_{v\in S}N(v)$ and $N[S]\coloneqq N(S)\cup S$. We define the private neighbor set $pn[v,S]$ of $v\in S$ as $pn[v,S]\coloneqq N[v] \setminus N[S\setminus\{v\}]$ and every element of $pn[v,S]$ is called a *private neighbor* of $v$. Note that a set $D\subseteq V$ is dominating if and only if $N[D]=V$. Further, $D$ is an mds if and only if additionally $pn[v,D]\neq\emptyset$ for all $v\in D$ holds. A *stable set* is a subset of pairwise non-adjacent vertices and a *clique* is a subset of pairwise adjacent vertices. Analogously to an mds, a stable set is a *maximal stable set* if it is not contained in another stable set. It is easy to see that every maximal stable set is an mds.

The length of a shortest path between two vertices $v,w\in V$ is denoted by $dist(v,w)$ and called the *distance* of $v$ and $w$. For two vertices $x,y\in V$ and a subset $V'\subseteq V$ with $x\in V',\, y\notin V'$ we write shortly $V'-x+y$ for $(V'\setminus \{x\})\cup \{y\}$. We use the abbreviation $[k]\coloneqq\{1,\ldots,k\}$ for a natural 