mirror of
https://github.com/janishutz/eth-summaries.git
synced 2025-11-25 10:34:23 +00:00
55 lines
2.5 KiB
TeX
55 lines
2.5 KiB
TeX
\newpage
|
|
\subsection{Connected Components}
|
|
\begin{definition}[]{Connected Component}
|
|
A \textbf{connected component} of a graph $G = (V, E)$ is a maximal subset of vertices $C \subseteq V$ such that:
|
|
\begin{itemize}
|
|
\item For every pair of vertices $u, v \in C$, there exists a path connecting $u$ and $v$.
|
|
\item Adding any vertex $w \notin C$ to $C$ would violate the connectedness condition.
|
|
\end{itemize}
|
|
\end{definition}
|
|
|
|
\begin{remarks}[]{Key Points About Connected Components}
|
|
\begin{itemize}
|
|
\item \textbf{Undirected Graphs:} A connected component is a subgraph where any two vertices are connected by a path, and which is connected to no additional vertices in the graph.
|
|
\item \textbf{Directed Graphs:} There are two types of connected components:
|
|
\begin{itemize}
|
|
\item \textbf{Strongly Connected Component:} A maximal subset of vertices where every vertex is reachable from every other vertex (considering edge direction).
|
|
\item \textbf{Weakly Connected Component:} A maximal subset of vertices where connectivity is considered by ignoring edge directions.
|
|
\end{itemize}
|
|
\end{itemize}
|
|
\end{remarks}
|
|
|
|
\begin{example}[]{Connected Components in a Graph}
|
|
\begin{center}
|
|
\begin{tikzpicture}[node distance=1.5cm, main/.style={circle, draw, fill=blue!20, minimum size=10mm, inner sep=0pt}]
|
|
% Connected component 1
|
|
\node[main] (1) {1};
|
|
\node[main] (2) [right of=1] {2};
|
|
\node[main] (3) [below of=1] {3};
|
|
\draw (1) -- (2);
|
|
\draw (1) -- (3);
|
|
\draw (2) -- (3);
|
|
|
|
% Connected component 2
|
|
\node[main] (4) [right of=2, xshift=2cm] {4};
|
|
\node[main] (5) [right of=4] {5};
|
|
\draw (4) -- (5);
|
|
|
|
% Isolated vertex (another component)
|
|
\node[main] (6) [below of=5, yshift=-1cm] {6};
|
|
\end{tikzpicture}
|
|
\end{center}
|
|
\end{example}
|
|
|
|
\begin{remarks}[]{Understanding the Example}
|
|
\begin{itemize}
|
|
\item In the given undirected graph:
|
|
\begin{itemize}
|
|
\item \textbf{Component 1:} $\{1, 2, 3\}$ (fully connected subgraph).
|
|
\item \textbf{Component 2:} $\{4, 5\}$ (connected by a single edge).
|
|
\item \textbf{Component 3:} $\{6\}$ (an isolated vertex).
|
|
\end{itemize}
|
|
\item These subsets are disjoint and collectively partition the graph's vertex set $V$.
|
|
\end{itemize}
|
|
\end{remarks}
|