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Never Gude 2026-04-17 14:40:08 +02:00
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\documentclass{ngexrcs}
\usepackage{hyperref}
\title{1. Übungsblatt}
\subject{Algorithmische Graphentheorie}
\author{Jasper Gude}
\begin{document}
\maketitle
\points[\qquad]{20}
\section{Spannbäume \& Breitensuche}
Sei $G = \tup{V, E}$ ein zusammenhängender Graph mit Kantengewichten $w: E \to \NN$
und $s \in V \deg$ ein ausgezeichneter Knoten.
\begin{enumerate}
\item
\begin{quote}
Wenn $w(e) = 1$ für alle $e \in E$, dann ist der Breitensuchbaum mit
Quelle $s$ ein minimaler Spannbaum.
\end{quote}
Die Breitensuche berechnet in diesem
Fall den kürzesten Weg von jedem Knoten zum Knoten $s$, also den
Breitensuchbaum. Dieser spannt also einen minimalen Spannbaum auf.
\points{2}
\item
\begin{quote}
Wenn $w(e) = 1$ für alle $e \in E$, dann ist jeder minimale Spannbaum
von $G$ ein Breitensuchbaum mit Quelle $s$.
\end{quote}
Falsch, siehe \autoref{fig:msb}.
\begin{figure}[h]
\centering
\includegraphics[width=0.2\textwidth]{msb.eps}
\caption{$\pi$-Zeiger des Breitensuchbaums und MSB blau hinterlegt.}
\label{fig:msb}
\end{figure}
\item
\begin{quote}
Wenn $w(e) \in \set{1, 2, 3}$ für alle $e \in E$, dann ist jeder minimale
Spannbaum von $G$ ein Tiefensuchbaum mit Quelle $s$.
\end{quote}
Sei $w(e) = 1$ für
alle $e \in E$ so gilt das Gegenbeispiel von oben. Also ist die Aussage
falsch.
\end{enumerate}
\section{Kreissuche}
\section{Eulerwege}
\section{Graphmodellierung}
\end{document}

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