From 896fb9601e19e7e0f3e79a6aac7c1448dcef7d54 Mon Sep 17 00:00:00 2001 From: Martin Thoma Date: Thu, 13 Jun 2013 11:18:18 +0200 Subject: [PATCH] =?UTF-8?q?Schlingen=20hinzugef=C3=BCgt;=20Textsetzungspro?= =?UTF-8?q?bleme=20behoben;=20Aufgabe=20'Zeichne=20alle=20Graphen'=20verbe?= =?UTF-8?q?ssert?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../Diskrete-Mathematik/Handout/Handout.tex | 29 +++++++++++------- .../LaTeX/Graphentheorie-I.pdf | Bin 1669608 -> 1682637 bytes .../Diskrete-Mathematik/LaTeX/Grundlagen.tex | 19 +++++++++++- .../LaTeX/graphs/graph-2-schlinge.tex | 29 ++++++++++++++++++ .../LaTeX/graphs/k-5-schlinge.tex | 28 +++++++++++++++++ 5 files changed, 93 insertions(+), 12 deletions(-) create mode 100644 presentations/Diskrete-Mathematik/LaTeX/graphs/graph-2-schlinge.tex create mode 100644 presentations/Diskrete-Mathematik/LaTeX/graphs/k-5-schlinge.tex diff --git a/presentations/Diskrete-Mathematik/Handout/Handout.tex b/presentations/Diskrete-Mathematik/Handout/Handout.tex index 7afc7f3..6f423d3 100644 --- a/presentations/Diskrete-Mathematik/Handout/Handout.tex +++ b/presentations/Diskrete-Mathematik/Handout/Handout.tex @@ -131,6 +131,12 @@ Sei $e \in E$ und $k = \Set{e_1, e_2} \in K$. $e$ heißt \textbf{inzident} zu $k :\Leftrightarrow e = e_1$ oder $e = e_2$ \end{definition} +\begin{definition}{Schlinge} +Sei $G=(E, K)$ ein Graph und $k=\Set{e_1, e_2} \in K$ eine Kante. + +$k$ heißt \textbf{Schlinge} $:\Leftrightarrow e_1 = e_2$ +\end{definition} + \begin{definition}{Vollständiger Graph} Sei $G = (E, K)$ ein Graph. @@ -210,7 +216,7 @@ $A$ heißt \textbf{eulerscher Kreis} $:\Leftrightarrow \forall_{k \in K}: k \in \begin{definition}{Eulerscher Graph} Ein Graph heißt \textbf{eulersch}, wenn er einen eulerschen Kreis enthält. \end{definition} - +\vspace{0.5cm} \begin{theorem}{Euler 1736} ~~~ \begin{precondition} @@ -224,14 +230,14 @@ Ein Graph heißt \textbf{eulersch}, wenn er einen eulerschen Kreis enthält. $\Rightarrow $ Es gilt: $\forall e \in E \exists i \in \Set{0, \dots, n}: e = e_i$ und alle Kanten aus $G$ sind genau ein mal in $C$.\\ Außerdem gilt: \[\text{Grad}(e_i) = \begin{cases} - 2 \cdot \text{Anzahl der vorkommen von } e_i \text{in } C & \text{falls } i\neq 0\\ - 2 \cdot (\text{Anzahl der vorkommen von } e_i -1) \text{in } C & \text{falls } i = 0\\ + 2 \cdot \text{Anzahl der Vorkommen von } e_i \text{ in } C & \text{falls } i\neq 0\\ + 2 \cdot (\text{Anzahl der Vorkommen von } e_i -1) \text{ in } C & \text{falls } i = 0\\ \end{cases} \] $\Rightarrow \forall e \in E: \text{Grad}(e)$ ist gerade \end{Proof} \end{theorem} - +\vspace{0.5cm} \begin{theorem}{Umkehrung des Satzes von Euler} ~~~ \begin{precondition} @@ -244,7 +250,7 @@ Ein Graph heißt \textbf{eulersch}, wenn er einen eulerschen Kreis enthält. \underline{I.A.:} $m=0$: $G$ ist eulersch. \cmark\\ $m=1$: Es gibt keinen Graphen in dem jede Ecke geraden Grad hat. \cmark\\ $m=2$: Nur ein Graph ist zusammenhängend, hat zwei Kanten und nur Ecken geraden Grades. Dieser ist eulersch. \cmark - + \goodbreak \underline{I.V.:} Sei $m \in \mathbb{N}_0$ beliebig, aber fest und es gelte: @@ -264,14 +270,15 @@ Ein Graph heißt \textbf{eulersch}, wenn er einen eulerschen Kreis enthält. Es gilt: \begin{itemize} \item Jede Ecke in $G^*$ hat geraden Grad - \item Jede Zusammenhangskomponente (Zshk) hat weniger als $m$ Knoten - \item[$\Rightarrow$] IV ist auf jede Zshk anwendbar - \item[$\Rightarrow$] Jede Zshk hat einen Eulerkreis + \item Jede Zusammenhangskomponente hat weniger als $m$ Knoten + \item[$\Rightarrow$] I.V. ist auf jede Zusammenhangskomponente anwendbar + \item[$\Rightarrow$] Jede Zusammenhangskomponente hat einen Eulerkreis \item[$\Rightarrow$] Man kann den Kreis $C$ durch die Eulerkreise erweitern und erhält insgesamt einen Eulerkreis \end{itemize} $\Rightarrow$ $G$ ist eulersch \end{Proof} \end{theorem} +\vspace{0.5cm} \begin{definition}{Offene eulersche Linie} Sei $G$ ein Graph und $A$ ein Weg, der kein Kreis ist. @@ -279,8 +286,8 @@ Sei $G$ ein Graph und $A$ ein Weg, der kein Kreis ist. $A$ heißt \textbf{offene eulersche Linie} von $G :\Leftrightarrow$ Jede Kante in $G$ kommt genau ein mal in $A$ vor. \end{definition} - -\begin{theorem}{Satz 8.2.3} +\vspace{0.5cm} +\begin{theorem}{} ~~~ \begin{precondition} Sei $G = (E, K)$ ein zusammenhängender Graph. @@ -296,7 +303,7 @@ in $G$ kommt genau ein mal in $A$ vor. \end{Proof} \end{theorem} - +\vfill Alle \LaTeX-Quellen und die neueste Version der PDF sind unter \href{https://github.com/MartinThoma/LaTeX-examples/tree/master/presentations/Diskrete-Mathematik}{github.com/MartinThoma/LaTeX-examples/tree/master/presentations/Diskrete-Mathematik} diff --git a/presentations/Diskrete-Mathematik/LaTeX/Graphentheorie-I.pdf b/presentations/Diskrete-Mathematik/LaTeX/Graphentheorie-I.pdf index c3ca7f702dc755097a8006b221d9db33b8d0d447..e723a518465756c430950b6deb0c6970a3b33329 100644 GIT binary patch delta 601221 zcmYIuV_+st({*gywr$(CZEmpf6>e-?vF(kMjqPM(+qUuU{d_;ZAJbDkU46Q$Yo@2` z%$(d?POVy@QmRNuF|smo!copH46VSivI0#}FwlQfFo1*dCaZ#{g0ZnApMWcXvNm!; zVA3Zia@zv87(n5{S=rbc`F$;eS(RWns_6Z^mbv7azbLATL4kX)99_}R$&U2G+WH*fq%TCqQ 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z+a86$bU>W?q7xXo6yWk5^lefQ#z4DODUl%hAv_wKAlSNulnEQ0hd#Qgh{fRO84>j+qdi26bVVVwaT$s+!tjEjZ%S#dM&4iIDXs%3& zPv+uC2d7c!XNN%1zUbx`UGATMChqaf_Pvpddh z(-bV4IuPjl`~hhtK4e5+9bN03+cqC+iiR-(cs22-MZ@>~@M^?Nk{LNa%DVB@2Utza zJf_0yJ+#mXookGkQcD1$*-U=Khx$`P$^jW_UK!ubW=`x_Rq=Ph>ds7{&W}(^$d9;D zM{M}uK7ZmLNagJLse7SPrfBd|&UC8)pU#9{bU>3D&I#Nu$s(CHKg%AmD9eXd#++d4 z1UVg|=)3$bWm(}I?UlKuonU$)6Z_NA4UEE{j# zlGaSlY}Cn*%j?vISo2xE{Er8Robht^gQWluAJ*Uy_cI~kXS`0ZDyk}LsHv;6?$}{y IZp8Zk0nBV(0RR91 diff --git a/presentations/Diskrete-Mathematik/LaTeX/Grundlagen.tex b/presentations/Diskrete-Mathematik/LaTeX/Grundlagen.tex index 2a7e27b..ef9470d 100644 --- a/presentations/Diskrete-Mathematik/LaTeX/Grundlagen.tex +++ b/presentations/Diskrete-Mathematik/LaTeX/Grundlagen.tex @@ -60,8 +60,25 @@ Hat eine Ecke den Grad 0, so nennt man ihn \textbf{isoliert}. \end{gallery} \end{frame} +\begin{frame}{Schlinge} +\begin{block}{Schlinge} +Sei $G=(E, K)$ ein Graph und $k=\Set{e_1, e_2} \in K$ eine Kante. + +$k$ heißt \textbf{Schlinge} $:\Leftrightarrow e_1 = e_2$ +\end{block} + +Ein Graph ohne Schlingen heißt \enquote{schlingenfrei} + +\begin{gallery} + \galleryimage{graphs/graph-1} + \galleryimage{graphs/graph-2-schlinge} + \galleryimage{graphs/k-3-3} + \galleryimage{graphs/k-5-schlinge} +\end{gallery} +\end{frame} + \begin{frame}{Aufgabe 1} -Zeichnen Sie alle Graphen mit genau vier Ecken. +Zeichnen Sie alle schlingenfreien Graphen mit genau vier Ecken. \only<2>{ \begin{gallery} diff --git a/presentations/Diskrete-Mathematik/LaTeX/graphs/graph-2-schlinge.tex b/presentations/Diskrete-Mathematik/LaTeX/graphs/graph-2-schlinge.tex new file mode 100644 index 0000000..41ddc97 --- /dev/null +++ b/presentations/Diskrete-Mathematik/LaTeX/graphs/graph-2-schlinge.tex @@ -0,0 +1,29 @@ +\documentclass[varwidth=true, border=2pt]{standalone} +\usepackage{tikz} +\usetikzlibrary{arrows,positioning} +\tikzset{ + %Define standard arrow tip + >=stealth', + % Define arrow style + pil/.style={->,thick} +} +\tikzstyle{vertex}=[draw,fill=black,circle,minimum size=10pt,inner sep=0pt] +\begin{document} + \begin{tikzpicture} + \node (a)[vertex] at (0,3) {}; + \node (b)[vertex] at (0,1) {}; + \node (c)[vertex] at (1,0) {}; + \node (d)[vertex] at (2,0) {}; + \node (e)[vertex] at (3,0) {}; + \node (f)[vertex] at (4,1) {}; + \node (g)[vertex] at (4,3) {}; + + \foreach \from/\to in {b/c,c/d,d/e,e/f} + \draw[line width=2pt] (\from) -- (\to); + + \path[line width=2pt] (d) edge[ out=140, in=50 + , looseness=0.8, loop + , distance=2cm] + node[above=3pt] {$k$} (d); + \end{tikzpicture} +\end{document} diff --git a/presentations/Diskrete-Mathematik/LaTeX/graphs/k-5-schlinge.tex b/presentations/Diskrete-Mathematik/LaTeX/graphs/k-5-schlinge.tex new file mode 100644 index 0000000..50adf14 --- /dev/null +++ b/presentations/Diskrete-Mathematik/LaTeX/graphs/k-5-schlinge.tex @@ -0,0 +1,28 @@ +% A complete graph +% Author: Quintin Jean-Noël +% +\documentclass[varwidth=true, border=2pt]{standalone} +\usepackage[nomessages]{fp}% http://ctan.org/pkg/fp +\usepackage{tikz} +\usetikzlibrary[topaths] + +\tikzstyle{vertex}=[draw,fill=black,circle,minimum size=10pt,inner sep=0pt] +\begin{document} + \newcommand\n{5} + \begin{tikzpicture} + %the multiplication with floats is not possible. Thus I split the loop in two. + \foreach \number in {1,...,\n}{ + \node[vertex] (N-\number) at ({\number*(360/\n)}:5.4cm) {}; + } + + \foreach \number in {1,...,\n}{ + \foreach \y in {1,...,\n}{ + \draw (N-\number) -- (N-\y); + } + \path (N-\number) edge[ out=140, in=50 + , looseness=0.8, loop + , distance=2cm] + node[above=3pt] {} (N-\number); + } + \end{tikzpicture} +\end{document}

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