From 55f4546a9f01a3300b9637c992fe5ae44ccff921 Mon Sep 17 00:00:00 2001 From: Valentin Brandl Date: Tue, 4 Dec 2018 15:20:35 +0100 Subject: [PATCH] Add stuff --- school/di-ma/uebung/07/07_1.tex | 84 ++++++++++++++++++++++++++++ school/di-ma/uebung/07/07_3.tex | 12 ++++ school/di-ma/uebung/07/07_3_b_1.jpg | Bin 0 -> 103580 bytes school/di-ma/uebung/07/07_3_b_2.jpg | Bin 0 -> 4384213 bytes school/di-ma/uebung/07/07_3_i.dot | 26 --------- 5 files changed, 96 insertions(+), 26 deletions(-) create mode 100644 school/di-ma/uebung/07/07_1.tex create mode 100644 school/di-ma/uebung/07/07_3_b_1.jpg create mode 100644 school/di-ma/uebung/07/07_3_b_2.jpg delete mode 100644 school/di-ma/uebung/07/07_3_i.dot diff --git a/school/di-ma/uebung/07/07_1.tex b/school/di-ma/uebung/07/07_1.tex new file mode 100644 index 0000000..0a2b38a --- /dev/null +++ b/school/di-ma/uebung/07/07_1.tex @@ -0,0 +1,84 @@ +\documentclass[12pt,a4paper,german]{article} +\usepackage{url} +%\usepackage{graphics} +\usepackage{times} +\usepackage[T1]{fontenc} +\usepackage{ngerman} +\usepackage{float} +\usepackage{diagbox} +\usepackage[utf8]{inputenc} +\usepackage{geometry} +\usepackage{amsfonts} +\usepackage{amsmath} +\usepackage{cancel} +\usepackage{wasysym} +\usepackage{csquotes} +\usepackage{graphicx} +\usepackage{epsfig} +\usepackage{paralist} +\usepackage{tikz} +\geometry{left=2.0cm,textwidth=17cm,top=3.5cm,textheight=23cm} + +%%%%%%%%%% 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 {7} % +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + % 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 7.1} + +Sei $G=(V,E)$ ein Graph. Mit $G^c := (V,(\frac{V}{2})\setminus E )$ bezeichnen wir den zu G komplementären Graphen. In Hausübung 6.3 wurde definiert, wann ein Graph asymmetrisch ist. Zeige: G ist asymmetrisch $\Leftrightarrow G^c$ ist asymmetrisch\\ +\\ +\noindent +Definition Asymmetrisch:\\ +Sei $G = (V,E)$ ein ungerichteter Graph. Ein Automorphismus von G ist ein Isomorphismus +von G auf sich selbst, d.h. eine Bijektion $f : V \rightarrow V$ , so dass $\{u,v\} \in E$ genau dann, wenn +$\{f(u),f(v)\}g \in E$. Ein Graph heißt genau dann asymmetrisch, wenn die Identität der einzige +Automorphismus ist.\\ +\\ +\noindent +\textbf{BEWEIS:}\\ +Seien $G=(V_1,E_1)$ und $G^c=(V_2,E_2)$ zwei Graphen, wobei $G^c$ der Komplementgraph von G ist.\\ +\\ +\noindent +G hat einen Automorphismus f.\\ +$\Leftrightarrow$\\ +Für alle Kanten $\{u,v\} \in E_1$ ist $\{f(u),f(v)\} \in E_1$ (also eine Kante und gleiches gilt umgekehrt).\\ +$\Leftrightarrow$\\ +Für alle Nichtkanten $\{u,v\} \not \in E_1$ ist $\{f(u),f(v)\} \not \in E_1$ (also keine Kante und gleiches gilt umgekehrt)\\ +$\Leftrightarrow$ (*)(**)\\ +Für alle Kanten $\{u,v\} \in E_2$ ist $\{f(u),f(v)\} \in E_2$ (also eine Kante und gleiches gilt umgekehrt)\\ +$\Leftrightarrow$\\ +Für alle Nichtkanten $\{u,v\} \not \in E_2$ ist $\{f(u),f(v)\} \not \in E_2$ (also keine Kante und gleiches gilt umgekehrt)\\ +$\Leftrightarrow$\\ +$G^c$ hat einen Automorphismus f.\\ +\\ +\noindent +G asymmetrisch dann ist \textbf{nur} die Identität ein Automorphismus von G und somit auch von $G^c$. +$G^c$ kann keinen weiteren Automorphismen haben, da dieser sonst einen Automorphismus auf G bildet.\\ +\\ +$G^c$ asymmetrisch dann ist \textbf{nur} die Identität ein Automorphismus von $G^c$ und somit auch von G. +G kann keinen weiteren Automorphismen haben, da dieser sonst einen Automorphismus auf $G^c$ bildet.\\ +\\ +(*) $\{(u,v)\}$ Kante in G $\Leftrightarrow$ $\{(u,v)\}$ Nichtkante in $G^c$\\ +(**) $\{(u,v)\}$ Nichtkante in G $\Leftrightarrow$ $\{(u,v)\}$ Kante in $G^c$ + +\end{document} diff --git a/school/di-ma/uebung/07/07_3.tex b/school/di-ma/uebung/07/07_3.tex index 4083183..e7801af 100644 --- a/school/di-ma/uebung/07/07_3.tex +++ b/school/di-ma/uebung/07/07_3.tex @@ -67,10 +67,22 @@ Hamiltonkreis: $(1,3,4,8,10,9,7,5,6,1)$ + \begin{figure}[h!] + \centering + \includegraphics[width=0.5\textwidth]{school/di-ma/uebung/07/07_3_b_1.jpg} + \caption{Entfernen eines inneren Knoten} + \end{figure} + \item Entefernen eines äußeren Knoten ($1$): Hamiltonkreis: $(2,7,9,3,4,5,6,10,8,2)$ + \begin{figure}[h!] + \centering + \includegraphics[width=0.5\textwidth]{school/di-ma/uebung/07/07_3_b_2.jpg} + \caption{Entfernen eines äußeren Knoten} + \end{figure} + \end{enumerate} Für alle (unterscheidbaren) Möglichkeiten, einen Knoten zu entfernen, wurde gezeigt, dass es einen diff --git a/school/di-ma/uebung/07/07_3_b_1.jpg b/school/di-ma/uebung/07/07_3_b_1.jpg new file mode 100644 index 0000000000000000000000000000000000000000..ff3b30e533112ebf6347b50217182d06b85e1874 GIT binary patch literal 103580 zcmbTdc~p|^7cTrTC}@s}C5E%F0!}kpqgD`TCZrDe`L&u~JL}O)Emwv@&wQ z0ddGiEq%+v%Avs_L@Tp0HOI`HbE?DdoU_hZ>-+2bK3M$4z2MpRv!8um*S@a(=l!1n zK!fDw=>~v6ARtu!2L5~mTmZPDB1};M4uipzl;FxL2vwC08&r@RwbT)Mx(51sx+s*P z=?-&4<83A=6nZyin}sC~hchrIIN+`9cUa@B{`(^!B_$=54JtaSsybFiC?l)?&#ylc zKohP2KqL^5F#y&CK{P>sUH}FF08)@&?SC)$f4@Lr2vk8423JzvAYV|Y0f0de2p9@c zP=G?^tCQs415izcjYgJEidue$Va5!E6(_qGZsL5mN!$PJPgCo#*c2t@O`CO)x__CO 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