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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}								%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 % DO NOT MODIFY THIS HEADER
\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 4.2}
\begin{enumerate}[1.]

    \item Graph $G=(V,E)$ mit\\
        Knotenmenge: $V=\{a,b,c,d,e,f\}$\\
        Kantenmenge: $E=
        \{
            \{a,b\},
            \{a,c\},
            \{a,d\},
            \{a,f\},
            \{b,f\},
            \{b,e\},
            \{c,d\},
            \{d,e\}
        \}
        $
        \\
        Adjazenz-Matrix:\\
        \begin{tabular}{c|ccccccc}
        & a & b & c & d & e &  f \\
        \hline
            a & 0 & 1 & 1 & 1 & 0 & 1 \\
            b & 1 & 0 & 0 & 0 & 1 & 1 \\
            c & 1 & 0 & 0 & 1 & 0 & 0 \\
            d & 1 & 0 & 1 & 0 & 1 & 0 \\
            e & 0 & 1 & 0 & 1 & 0 & 0 \\
            f & 1 & 1 & 0 & 0 & 0 & 0 \\
        \end{tabular}
        \\
        \\
        Inzidenz-Matrix:\\
        \begin{tabular}{c|ccccccccc}
            a & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\
            b & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 \\
            c & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 \\
            d & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 \\
            e & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 \\
            f & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 \\
        \end{tabular}

    \item \begin{tikzpicture}
        \def \n {5}
        \def \radius {3cm}
        \def \margin {8} % margin in angles, depends on the radius
        \node[draw, circle] (a) at (3,2) {a};
        \node[draw, circle] (b)  at (1,1) {b};
        \node[draw, circle] (c) at (2,1) {c};
        \node[draw, circle] (d)  at (3,1) {d};
        \node[draw, circle] (f)  at (4,1) {f};
        \node[draw, circle] (e)  at (0,0) {e};

        \draw (a)--(b);
        \draw (a)--(c);
        \draw (a)--(d);
        \draw (a)--(f);
        \draw (b)--(e);
        \end{tikzpicture}
        \\
        $G \setminus \{\{b,f\},\{c,d\},\{d,e\}\}$

        \begin{tikzpicture}
        \def \n {5}
        \def \radius {3cm}
        \def \margin {8} % margin in angles, depends on the radius
        \node[draw, circle] (b) at (2,2) {b};
        \node[draw, circle] (a)  at (1,1) {a};
        \node[draw, circle] (f) at (2,1) {f};
        \node[draw, circle] (e)  at (3,1) {e};
        \node[draw, circle] (c)  at (0,0) {c};
        \node[draw, circle] (d)  at (2,0) {d};

        \draw (b)--(a);
        \draw (b)--(f);
        \draw (b)--(e);
        \draw (a)--(c);
        \draw (a)--(d);

        \end{tikzpicture}
        \\
        $G \setminus \{\{a,f\},\{c,d\},\{d,e\}\}$

\end{enumerate}

\end{document}