Commit 708996f4 by msurl

 \section{Methods} \raggedbottom (Given an undirected Graph $G = (V,E)$.)?\\ (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: $\forall v \in V \setminus DS: \exists u \in DS, u \in N_k(v)$ \end{definition} \begin{definition}[connected k-hop Dominating Set] A $k$-hop Dominating Set $DS$ is a connected $k$-hop Dominating Set if the induced subgraph $G[DS]$ is connected. \end{definition} \begin{definition}[rooted connected k-hop Dominating Set] Let $v_{root} \in V$ be a firmly defined vertex. A rooted connected $k$-hop Dominating Set DS is as connected $k$-hop Dominating Set which also includes $v_{root}$. \end{definition}