Commit 708996f4 authored by msurl's avatar msurl
Browse files

added (first draft of) paragraph about methods and definitions

parent 1f5a512e
\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}
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