69 lines
2.0 KiB
TeX
69 lines
2.0 KiB
TeX
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\documentclass[12pt,a4paper,german]{article}
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\usepackage{url}
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%\usepackage{graphics}
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\usepackage{times}
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\usepackage[T1]{fontenc}
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\usepackage{ngerman}
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\usepackage{float}
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\usepackage{diagbox}
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\usepackage[utf8]{inputenc}
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\usepackage{geometry}
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\usepackage{amsfonts}
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\usepackage{amsmath}
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\usepackage{cancel}
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\usepackage{wasysym}
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\usepackage{csquotes}
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\usepackage{graphicx}
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\usepackage{epsfig}
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\usepackage{paralist}
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\usepackage{tikz}
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\geometry{left=2.0cm,textwidth=17cm,top=3.5cm,textheight=23cm}
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%%%%%%%%%% Fill out the the definitions %%%%%%%%%
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\def \name {Valentin Brandl} %
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\def \matrikel {108018274494} %
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\def \pname {Marvin Herrmann} %
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\def \pmatrikel {108018265436} %
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\def \gruppe {2 (Mi 10-12 Andre)}
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\def \qname {Pascal Brackmann}
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\def \qmatrikel {108017113834} %
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\def \uebung {9} %
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% DO NOT MODIFY THIS HEADER
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\newcommand{\hwsol}{
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\vspace*{-2cm}
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\noindent \matrikel \quad \name \hfill \"Ubungsgruppe: \gruppe \\
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\noindent \pmatrikel \quad \pname \\
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\noindent \qmatrikel \quad \qname \\
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\begin{center}{\Large \bf L\"osung f\"ur \"Ubung \# \uebung}\end{center}
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}
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\begin{document}
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%Import header
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\hwsol
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\section*{Aufgabe 9.1}
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\begin{enumerate}[1.]
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\item
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\includegraphics[width=\textwidth]{build/school/di-ma/uebung/09/09_1.jpg}
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\item $M_{perfekt} = \{\{1,5\}, \{2,7\}, \{3,8\}, \{4,6\}\}$
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Es gibt kein weiteres perfektes Matching
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$5$ und $8$ haben Grad 1, daher müssen die Kanten $\{1,5\}$ und $\{3,8\}$ im perfekten Matching vorkommen.
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$\{1,6\}$ und $\{3,6\}$ können nicht mehr verwendet werden, da die Knoten $1$ und $3$ schon im Matching
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vorkommen, also ist die einzige Kante, die von $6$ ausgehend im Matching vorkommt, $\{4,6\}$ sein.
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$1,3,4,5,6,8$ kommen schon im Matching vor, es fehlen also nur noch $2$ und $7$ und damit die Kante $\{2,7\}$
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\item $M_{maximal \land \lnot maximum} = \{\{1,4\}, \{2,7\}, \{3,6\}\}$
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\end{enumerate}
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\end{document}
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