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some changes (Knoten -> Ecken)
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Martin Thoma
parent
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commit
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12 changed files with 239 additions and 58 deletions
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@ -1,22 +1,22 @@
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\subsection{Grundlagen}
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\begin{frame}{Graph}
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\begin{block}{Graph}
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Ein Graph ist ein Tupel $(V, E)$, wobei $V \neq \emptyset$ die Knotenmenge und
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$E \subseteq V \times V$ die
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Ein Graph ist ein Tupel $(E, K)$, wobei $E \neq \emptyset$ die Eckenmenge und
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$K \subseteq E \times E$ die
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Kantenmenge bezeichnet.
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\end{block}
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\pause
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\tikzstyle{vertex}=[draw,fill=black,circle,minimum size=10pt,inner sep=0pt]
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\begin{gallery}
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\galleryimage{graphs/graph-1}
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\galleryimage{graphs/graph-2}
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\galleryimage{graphs/k-3-3}
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\galleryimage{graphs/k-5}\\
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\galleryimage{graphs/k-16}
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\galleryimage{graphs/graph-6}
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\galleryimage{graphs/star-graph}
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\galleryimage{graphs/tree}
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\galleryimage[Green]{graphs/graph-1}
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\galleryimage[Green]{graphs/graph-2}
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\galleryimage[Green]{graphs/k-3-3}
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\galleryimage[Green]{graphs/k-5}\\
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\galleryimage[Green]{graphs/k-16}
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\galleryimage[Green]{graphs/graph-6}
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\galleryimage[Green]{graphs/star-graph}
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\galleryimage[Green]{graphs/tree}
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\end{gallery}
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\end{frame}
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@ -30,54 +30,54 @@ Kantenmenge bezeichnet.
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\begin{frame}{Inzidenz}
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\begin{block}{Inzidenz}
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Sei $v \in V$ und $e = \Set{v_1, v_2} \in E$.
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Sei $e \in E$ und $k = \Set{v_1, v_2} \in K$.
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$v$ heißt \textbf{inzident} zu $e :\Leftrightarrow v = v_1$ oder $v = v_2$
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$e$ heißt \textbf{inzident} zu $k :\Leftrightarrow e = e_1$ oder $e = e_2$
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\end{block}
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\pause
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\tikzstyle{vertex}=[draw,fill=black,circle,minimum size=10pt,inner sep=0pt]
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\begin{gallery}
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\galleryimage{inzidenz/graph-1}
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\galleryimage{inzidenz/graph-2}
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\galleryimage{inzidenz/k-3-3}
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\galleryimage{inzidenz/k-5}\\
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\galleryimage{inzidenz/k-16}
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\galleryimage{inzidenz/graph-6}
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\galleryimage{inzidenz/star-graph}
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\galleryimage{inzidenz/tree}
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\galleryimage[Green]{inzidenz/graph-1}
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\galleryimage[Green]{inzidenz/graph-2}
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\galleryimage[Green]{inzidenz/k-3-3}
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\galleryimage[Green]{inzidenz/k-5}\\
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\galleryimage[Green]{inzidenz/k-16}
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\galleryimage[red]{inzidenz/graph-6}
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\galleryimage[Green]{inzidenz/star-graph}
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\galleryimage[Green]{inzidenz/tree}
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\end{gallery}
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\end{frame}
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\begin{frame}{Vollständige Graphen}
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\begin{block}{Vollständiger Graph}
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Sei $G = (V, E)$ ein Graph.
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Sei $G = (E, K)$ ein Graph.
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$G$ heißt \textbf{vollständig} $:\Leftrightarrow E = V \times V \setminus \Set{v \in V: \Set{v, v}}$
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$G$ heißt \textbf{vollständig} $:\Leftrightarrow = E \times E \setminus \Set{e \in E: \Set{e, e}}$
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\end{block}
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Ein vollständiger Graph mit $n$ Knoten wird als $K_n$ bezeichnet.
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Ein vollständiger Graph mit $n$ Ecken wird als $K_n$ bezeichnet.
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\pause
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\tikzstyle{vertex}=[draw,fill=black,circle,minimum size=10pt,inner sep=0pt]
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\begin{gallery}
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\galleryimage{vollstaendig/k-1}
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\galleryimage{vollstaendig/k-2}
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\galleryimage{vollstaendig/k-3}
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\galleryimage{vollstaendig/k-4}\\
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\galleryimage{vollstaendig/k-5}
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\galleryimage{vollstaendig/k-6}
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\galleryimage{vollstaendig/k-7}
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\galleryimage{vollstaendig/k-16}
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\galleryimage[Green]{vollstaendig/k-1}
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\galleryimage[Green]{vollstaendig/k-2}
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\galleryimage[Green]{vollstaendig/k-3}
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\galleryimage[Green]{vollstaendig/k-4}\\
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\galleryimage[Green]{vollstaendig/k-5}
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\galleryimage[Green]{vollstaendig/k-6}
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\galleryimage[Green]{vollstaendig/k-7}
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\galleryimage[Green]{vollstaendig/k-16}
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\end{gallery}
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\end{frame}
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\begin{frame}{Bipartite Graphen}
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\begin{block}{Bipartite Graph}
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Sei $G = (V, E)$ ein Graph und $A, B \subset V$ zwei disjunkte Knotenmengen mit
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$V \setminus A = B$.
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Sei $G = (E, K)$ ein Graph und $A, B \subset V$ zwei disjunkte Eckenmengen mit
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$E \setminus A = B$.
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$G$ heißt \textbf{bipartit} $:\Leftrightarrow \forall_{e = \Set{v_1, v_2} \in E}: (v_1 \in A \text{ und } v_2 \in B) \text{ oder } (v_1 \in B \text{ und } v_2 \in A) $
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$G$ heißt \textbf{bipartit} $:\Leftrightarrow \forall_{k = \Set{e_1, e_2} \in K}: (e_1 \in A \text{ und } e_2 \in B) \text{ oder } (e_1 \in B \text{ und } e_2 \in A) $
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\end{block}
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TODO: 8 Bilder von Graphen
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@ -85,40 +85,90 @@ TODO: 8 Bilder von Graphen
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\begin{frame}{Vollständig bipartite Graphen}
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\begin{block}{Vollständig bipartite Graphen}
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Sei $G = (V, E)$ ein bipartiter Graph und $\Set{A, B}$ bezeichne die Bipartition.
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Sei $G = (E, K)$ ein bipartiter Graph und $\Set{A, B}$ bezeichne die Bipartition.
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$G$ heißt \textbf{vollständig bipartit} $:\Leftrightarrow \forall_{a \in A} \forall_{b \in B}: \Set{a, b} \in E$
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$G$ heißt \textbf{vollständig bipartit} $:\Leftrightarrow \forall_{a \in A} \forall_{b \in B}: \Set{a, b} \in K$
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\end{block}
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TODO: 8 Bilder von Graphen
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\begin{gallery}
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\galleryimage[Green]{vollstaendig-bipartit/k-2-2}
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\galleryimage[Green]{vollstaendig-bipartit/k-2-3}
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\galleryimage[Green]{vollstaendig-bipartit/k-2-5}
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\galleryimage[Green]{vollstaendig-bipartit/k-3-3}\\
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\galleryimage[Green]{vollstaendig-bipartit/k-3-4}
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\galleryimage[Green]{vollstaendig-bipartit/k-3-5}
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\galleryimage[Green]{vollstaendig-bipartit/k-4-5}
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\galleryimage[Green]{vollstaendig-bipartit/k-5-5}
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\end{gallery}
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\end{frame}
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\begin{frame}{Vollständig bipartite Graphen}
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Bezeichnung: Vollständig bipartite Graphen mit der Bipartition $\Set{A, B}$
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bezeichnet man mit $K_{|A|, |B|}$.
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TODO: $K_{2,2}$
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TODO: $K_{2,3}$
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TODO: $K_{3,3}$
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\begin{gallery}
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\galleryimage[Green]{vollstaendig-bipartit/k-2-2}
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\galleryimage[Green]{vollstaendig-bipartit/k-2-3}
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\galleryimage[Green]{vollstaendig-bipartit/k-2-5}
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\galleryimage[Green]{vollstaendig-bipartit/k-3-3}\\
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\galleryimage[Green]{vollstaendig-bipartit/k-3-4}
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\galleryimage[Green]{vollstaendig-bipartit/k-3-5}
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\galleryimage[Green]{vollstaendig-bipartit/k-4-5}
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\galleryimage[Green]{vollstaendig-bipartit/k-5-5}
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\end{gallery}
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\end{frame}
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\begin{frame}{Kantenzug}
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\begin{block}{Kantenzug}
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Sei $G = (V, E)$ ein Graph.
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Sei $G = (E, K)$ ein Graph.
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Dann heißt eine Folge $e_1, e_2, \dots, e_s$ von Kanten, zu denen es Knoten
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$v_0, v_1, v_2, \dots, v_s$ gibt, so dass
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Dann heißt eine Folge $k_1, k_2, \dots, k_s$ von Kanten, zu denen es Ecken
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$e_0, e_1, e_2, \dots, e_s$ gibt, so dass
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\begin{itemize}
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\item $e_1 = \Set{v_0, v_1}$
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\item $e_2 = \Set{v_1, v_2}$
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\item $k_1 = \Set{e_0, e_1}$
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\item $k_2 = \Set{e_1, e_2}$
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\item \dots
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\item $e_s = \Set{v_{s-1}, v_s}$
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\item $k_s = \Set{e_{s-1}, e_s}$
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\end{itemize}
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gilt ein \textbf{Kantenzug}, der $v_0$ und $v_s$ \textbf{verbindet} und $s$
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gilt ein \textbf{Kantenzug}, der \textcolor{purple}{$e_0$} und \textcolor{blue}{$e_s$} \textbf{verbindet} und $s$
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seine \textbf{Länge}.
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\end{block}
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TODO: 8 Bilder
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\tikzstyle{vertex}=[draw,fill=black,circle,minimum size=10pt,inner sep=0pt]
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\adjustbox{max size={\textwidth}{0.2\textheight}}{
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\begin{tikzpicture}
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\node (a)[vertex] at (1,1) {};
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\node (b)[vertex] at (2,5) {};
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\node (c)[vertex] at (3,3) {};
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\node (d)[vertex] at (5,4) {};
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\node (e)[vertex] at (3,6) {};
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\node (f)[vertex] at (5,6) {};
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\node (g)[vertex] at (7,6) {};
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\node (h)[vertex] at (7,4) {};
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\node (i)[vertex] at (6,2) {};
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\node (j)[vertex] at (8,7) {};
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\node (k)[vertex] at (9,5) {};
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\node (l)[vertex] at (13,6) {};
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\node (m)[vertex] at (11,7) {};
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\node (n)[vertex] at (15,7) {};
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\node (o)[vertex] at (16,4) {};
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\node (p)[vertex] at (10,2) {};
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\node (q)[vertex] at (13,1) {};
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\node (r)[vertex] at (16,1) {};
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\node (s)[vertex] at (17,4) {};
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\node (t)[vertex] at (19,6) {};
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\node (u)[vertex] at (18,3) {};
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\node (v)[vertex] at (20,2) {};
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\node (w)[vertex] at (15,4) {};
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\foreach \from/\to in {a/c,c/b,c/d,d/f,f/g,g/h,h/d,d/g,h/f,i/k,k/j,k/l,l/m,m/n,n/o,o/t,t/v,v/u,s/r,o/q,q/p,u/t}
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\draw[line width=2pt] (\from) -- (\to);
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\node (i)[vertex,purple] at (6,2) {};
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\node (v)[vertex,blue] at (20,2) {};
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\draw[line width=4pt, red] (i) -- (k) -- (l) -- (m) -- (n) -- (o) -- (t) -- (v);
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\end{tikzpicture}
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}
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\end{frame}
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\begin{frame}{Geschlossener Kantenzug}
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@ -161,14 +211,14 @@ $G$ heißt \textbf{zusammenhängend} $:\Leftrightarrow \forall v_1, v_2 \in V: $
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TODO: 8 Bilder
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\end{frame}
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\begin{frame}{Grad eines Knotens}
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\begin{block}{Grad eines Knotens}
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Der \textbf{Grad} eines Knotens ist die Anzahl der Kanten, die von diesem Knoten
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\begin{frame}{Grad einer Ecke}
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\begin{block}{Grad einer Ecke}
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Der \textbf{Grad} einer Ecke ist die Anzahl der Kanten, die von dieser Ecke
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ausgehen.
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\end{block}
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\begin{block}{Isolierte Knoten}
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Hat ein Knoten den Grad 0, so nennt man ihn \textbf{isoliert}.
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\begin{block}{Isolierte Ecken}
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Hat eine Ecke den Grad 0, so nennt man ihn \textbf{isoliert}.
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\end{block}
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TODO: 8 Bilder
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@ -15,8 +15,9 @@
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\node[vertex] (N-\number) at ({\number*(360/\n)}:5.4cm) {};
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}
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\node[vertex] (N-3) at ({3*(360/\n)}:5.4cm) {};
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\node[vertex] (N-4) at ({4*(360/\n)}:5.4cm) {};
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\node[vertex,red] (N-1) at ({1*(360/\n)}:5.4cm) {};
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\node[vertex,red] (N-2) at ({2*(360/\n)}:5.4cm) {};
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\node[vertex,red] (N-5) at ({5*(360/\n)}:5.4cm) {};
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\draw (N-1) -- (N-2);
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\draw (N-2) -- (N-3);
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@ -0,0 +1,16 @@
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\documentclass[varwidth=true, border=2pt]{standalone}
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\usepackage{tikz}
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\usetikzlibrary{arrows,positioning}
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\begin{document}
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\tikzstyle{vertexs}=[draw,fill=black,circle,minimum size=4pt,inner sep=0pt]
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\begin{tikzpicture}
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\foreach \x in {0,1}
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\foreach \y in {0,1}{
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\node (a)[vertexs] at (\y,0) {};
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\node (b)[vertexs] at (\x,1) {};
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\draw (a) -- (b);
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}
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\end{tikzpicture}
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\end{document}
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@ -0,0 +1,16 @@
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\documentclass[varwidth=true, border=2pt]{standalone}
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\usepackage{tikz}
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\usetikzlibrary{arrows,positioning}
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\begin{document}
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\tikzstyle{vertexs}=[draw,fill=black,circle,minimum size=4pt,inner sep=0pt]
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\begin{tikzpicture}
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\foreach \x in {0,1}
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\foreach \y in {0,1,2}{
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\node (a)[vertexs] at (\y,0) {};
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\node (b)[vertexs] at (\x,1) {};
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\draw (a) -- (b);
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}
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\end{tikzpicture}
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\end{document}
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@ -0,0 +1,16 @@
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\documentclass[varwidth=true, border=2pt]{standalone}
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\usepackage{tikz}
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\usetikzlibrary{arrows,positioning}
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\begin{document}
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\tikzstyle{vertexs}=[draw,fill=black,circle,minimum size=4pt,inner sep=0pt]
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\begin{tikzpicture}
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\foreach \x in {0,1}
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\foreach \y in {0,1,2,3,4}{
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\node (a)[vertexs] at (\y,0) {};
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\node (b)[vertexs] at (\x,1) {};
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\draw (a) -- (b);
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}
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\end{tikzpicture}
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\end{document}
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@ -0,0 +1,16 @@
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\documentclass[varwidth=true, border=2pt]{standalone}
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\usepackage{tikz}
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\usetikzlibrary{arrows,positioning}
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\begin{document}
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\tikzstyle{vertexs}=[draw,fill=black,circle,minimum size=4pt,inner sep=0pt]
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\begin{tikzpicture}
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\foreach \x in {0,1,2}
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\foreach \y in {0,1,2}{
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\node (a)[vertexs] at (\y,0) {};
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\node (b)[vertexs] at (\x,1) {};
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\draw (a) -- (b);
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}
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\end{tikzpicture}
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\end{document}
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@ -0,0 +1,16 @@
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\documentclass[varwidth=true, border=2pt]{standalone}
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\usepackage{tikz}
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\usetikzlibrary{arrows,positioning}
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\begin{document}
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\tikzstyle{vertexs}=[draw,fill=black,circle,minimum size=4pt,inner sep=0pt]
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\begin{tikzpicture}
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\foreach \x in {0,1,2}
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\foreach \y in {0,1,2,3}{
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\node (a)[vertexs] at (\y,0) {};
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\node (b)[vertexs] at (\x,1) {};
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\draw (a) -- (b);
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}
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\end{tikzpicture}
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\end{document}
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@ -0,0 +1,16 @@
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\documentclass[varwidth=true, border=2pt]{standalone}
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\usepackage{tikz}
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\usetikzlibrary{arrows,positioning}
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\begin{document}
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\tikzstyle{vertexs}=[draw,fill=black,circle,minimum size=4pt,inner sep=0pt]
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\begin{tikzpicture}
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\foreach \x in {0,1,2}
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\foreach \y in {0,1,2,3,4}{
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\node (a)[vertexs] at (\y,0) {};
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\node (b)[vertexs] at (\x,1) {};
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\draw (a) -- (b);
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}
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\end{tikzpicture}
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\end{document}
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@ -0,0 +1,16 @@
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\documentclass[varwidth=true, border=2pt]{standalone}
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\usepackage{tikz}
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\usetikzlibrary{arrows,positioning}
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\begin{document}
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\tikzstyle{vertexs}=[draw,fill=black,circle,minimum size=4pt,inner sep=0pt]
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\begin{tikzpicture}
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\foreach \x in {0,1,2,3}
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\foreach \y in {0,1,2,3,4}{
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\node (a)[vertexs] at (\y,0) {};
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\node (b)[vertexs] at (\x,1) {};
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\draw (a) -- (b);
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}
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\end{tikzpicture}
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\end{document}
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@ -0,0 +1,16 @@
|
|||
\documentclass[varwidth=true, border=2pt]{standalone}
|
||||
\usepackage{tikz}
|
||||
\usetikzlibrary{arrows,positioning}
|
||||
|
||||
\begin{document}
|
||||
\tikzstyle{vertexs}=[draw,fill=black,circle,minimum size=4pt,inner sep=0pt]
|
||||
|
||||
\begin{tikzpicture}
|
||||
\foreach \x in {0,1,2,3,4}
|
||||
\foreach \y in {0,1,2,3,4}{
|
||||
\node (a)[vertexs] at (\y,0) {};
|
||||
\node (b)[vertexs] at (\x,1) {};
|
||||
\draw (a) -- (b);
|
||||
}
|
||||
\end{tikzpicture}
|
||||
\end{document}
|
||||
Loading…
Add table
Add a link
Reference in a new issue