(May describe what this Graph represents at this stage)\\

(Maybe avoid too much formalism?)

\begin{definition}[Neighborhood]

Given an undirected Graph $G =(V,E)$. Let $N(v)$ denote the neighborhood of a vertex $v$. $N(v)$ can formally be described as follows: \[w \in N(v)\Leftrightarrow\exists(v,w)\in E\]

\end{definition}

(Maybe leaf base case out as the s.t. part ist different from k-hop version. So after introducing in the implementation part it would be replaced.)

\begin{definition}[Dominating Set]

Given an undirected Graph $G =(V,E)$ a Dominating Set is a subset $DS \subset V$ such that each vertex $v \in V$ is either included in the Dominating Set or adjacent to at least one vertex which is included in the Dominating Set. So a for a Dominating Set $DS$ the following statement is valid

\[\forall v \in V \setminus DS: \exists u \in DS, u \in N(v)\]

\end{definition}

\begin{definition}[k-Neighborhood]

The neighborhood of a single vertex $N(v)$ is defined above. Let the neighborhood of a set of vertices $W \subset V$ be defined as follows:

\[N(W) :=\bigcup_{u \in W} N(u)\]

Let $k \in\mathbb{N}$.

With help of this definition the k-neighborhood $N_k(v)$ of a single vertex $v \in V$ can recursively be defined as:

\[N_k(v) := N(N_{k-1}(v))\setminus v\]

wheras $N_1(v)= N(v)$. So $N_k(v)$ is a set of all vertices which can be reached with at most $k$ steps starting from $v$.

\end{definition}

\begin{definition}[k-hop Dominating Set]

A $k$-hop Dominating Set is a subset $DS \subset V$ such that for each vertex $v \in V \setminus DS$ there exists a path of length $l \leq k$ between $v$ and at least one vertex $d \in DS$. So $DS$ is a $k$-hop dominating set if it fulfills the following requirement:

\[\forall v \in V \setminus DS: \exists u \in DS, u \in N_k(v)\]