notes/school/di-ma/uebung/04/04_3.tex

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%%%%%%%%%% Fill out the the definitions %%%%%%%%%
\def \name {Valentin Brandl}					%
\def \matrikel {108018274494}				%
\def \pname {Marvin Herrmann}				%
\def \pmatrikel {108018265436}				%
\def \gruppe {2 (Mi 10-12 Andre)}
\def \qname {Pascal Brackmann}
\def \qmatrikel {108017113834}				%
\def \uebung {4}								%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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\newcommand{\hwsol}{
\vspace*{-2cm}
\noindent \matrikel \quad \name \hfill \"Ubungsgruppe: \gruppe \\
\noindent \pmatrikel \quad \pname \\
\noindent \qmatrikel \quad \qname \\
\begin{center}{\Large \bf L\"osung f\"ur \"Ubung \# \uebung}\end{center}
}

\begin{document}
%Import header
\hwsol

\section*{Aufgabe 4.3}
\begin{enumerate}[1.]

    \item Graph $G=(V,E)$ mit\\
        Knotenmenge: $V=\{1,2,3,4,5,6\}$\\
        Kantenmenge: $E=
        \{
            (1,2),
            (1,3),
            (2,3),
            (3,4),
            (4,5),
            (5,6),
            (6,4)
        \}$
        \\

        Adjazenz-Matrix:\\
        \begin{tabular}{c|cccccc}
        & 1 &  2 & 3 & 4 & 5 & 6 \\
        \hline
            1 & 0 & 1 & 1 & 0 & 0& 0 \\
            2 & 0 & 0 & 1 & 0 & 0& 0 \\
            3 & 0 & 0 & 0 & 1 & 0& 0 \\
            4 & 0 & 0 & 0 & 0 & 1& 0 \\
            5 & 0 & 0 & 0 & 0 & 0& 1 \\
            6 & 0 & 0 & 0 & 1 & 0& 0 \\
        \end{tabular}
        \\
        Inzidenz-Matrix:\\
        \begin{tabular}{c|ccccccc}
            1 & -1 & -1 & 0 & 0 & 0 & 0 & 0 \\
            2 & 1 & 0 & -1 & 0 & 0 & 0 & 0 \\
            3 & 0 & 1 & 1 & -1 & 0 & 0 & 0 \\
            4 & 0 & 0 & 0 & 1 & -1 & 0 & 1 \\
            5 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \\
            6 & 0 & 0 & 0 & 0 & 0 & 1 & -1 \\
        \end{tabular}

    \item \begin{tikzpicture}
            \def \n {5}
            \def \radius {3cm}
            \def \margin {8} % margin in angles, depends on the radius
            \node[draw, circle] (1) at (0,1) {1};
            \node[draw, circle] (4) at (2,2) {4};
            \node[draw, circle] (6)  at (2,0) {6};
            \node[draw, circle] (3)  at (1,0) {3};
            \draw [->] 
            (1) edge (3)
            (3) edge (4)
            (6) edge (4);
        \end{tikzpicture}
        \\
        $V'=\{1,3,4,6\}$

    \item \begin{tikzpicture}
            \def \n {5}
            \def \radius {3cm}
            \def \margin {8} % margin in angles, depends on the radius
            \node[draw, circle] (1) at (0,0) {1};
            \node[draw, circle] (3) at (1,0) {3};
            \node[draw, circle] (4)  at (2,0) {4};
            \node[draw, circle] (5)  at (3,0) {5};
            \draw [->] 
            (1) edge (3)
            (3) edge (4)
            (4) edge (5);
        \end{tikzpicture}
        \\
        $V'=\{1,3,4,5\}$

\end{enumerate}

\end{document}