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Inoffizielles Skript.lyx
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Inoffizielles Skript.lyx
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#LyX 2.2 created this file. For more info see http://www.lyx.org/
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eqs-within-sections
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algorithm2e
enumitem
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\index Index
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\end_header
\begin_body
\begin_layout Title
Algorithmen für planaren Graphen
\end_layout
\begin_layout Subtitle
Mitschrieb der Vorlesung von Prof.
Dr.
Dorothea Wagner am KIT im Sommersemester 2016
\end_layout
\begin_layout Author
Joshua Gleitze
\end_layout
\begin_layout Address
Dieses Skript wird in einem GitHub-Repository unter
\begin_inset CommandInset href
LatexCommand href
target "https://github.com/jGleitz/AfpG-Skript"
\end_inset
entwickelt.
Es ist noch nicht fertig und enthält sicherlich noch Fehler.
Jeder Beitrag (auch nur die Information über einen Fehler) ist herzlich
willkommen!
\end_layout
\begin_layout Standard
\begin_inset FormulaMacro
\renewcommand{\O}[1]{\mathcal{O}\left(#1\right)}
{\mathcal{O}\left(#1\right)}
\end_inset
\begin_inset FormulaMacro
\newcommand{\dist}[1]{\text{dist}\left(#1\right)}
{\text{dist}\left(#1\right)}
\end_inset
\end_layout
\begin_layout Standard
\begin_inset FormulaMacro
\newcommand{\NP}{\mathcal{NP}}
{\mathcal{NP}}
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
pagenumbering{gobble}
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset CommandInset toc
LatexCommand tableofcontents
\end_inset
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\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
listofpropositions
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status open
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\backslash
clearpage
\backslash
pagenumbering{arabic}
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\end_inset
\end_layout
\begin_layout Chapter
Einführung
\end_layout
\begin_layout Standard
Diese Vorlesung handelt meist von einem ungerichteten, einfachen Graph
\begin_inset Formula $G=\left(V,E\right)$
\end_inset
ohne Schlingen.
Kann man
\begin_inset Formula $G$
\end_inset
so zeichnen, dass sich keine seiner Kanten kreuzen, so bezeichnet man
\begin_inset Formula $G$
\end_inset
als planar.
Dies wird im Folgenden formal definiert.
\end_layout
\begin_layout Standard
Prinzipiell kann ein Graph
\begin_inset Formula $G=\left(V,E\right)$
\end_inset
dargestellt werden indem man seine Knoten
\begin_inset Formula $v\in V$
\end_inset
auf Punkte in
\begin_inset Formula $\mathbb{R}^{2}$
\end_inset
und seine Kanten
\begin_inset Formula $e\in E$
\end_inset
auf Jordan-Kurven in
\begin_inset Formula $\mathbb{R}^{2}$
\end_inset
, die die entsprechenden Punkte verbinden, abbildet.
\end_layout
\begin_layout Definition
\begin_inset Index idx
status open
\begin_layout Plain Layout
Jordan-Kurve
\end_layout
\end_inset
Eine stetige, sich nicht selbst kreuzende Kurve heißt
\noun on
Jordan-Kurve
\noun default
.
\end_layout
\begin_layout Standard
\begin_inset Separator plain
\end_inset
\end_layout
\begin_layout Notation*
\begin_inset Index idx
status open
\begin_layout Plain Layout
\begin_inset Formula $K_{n}$
\end_inset
\end_layout
\end_inset
\begin_inset Formula $K_{n}=\left(V,E\right)$
\end_inset
bezeichnet den vollständigen Graph mit
\begin_inset Formula $\left|V\right|=n$
\end_inset
und
\begin_inset Formula
\[
E=\left\{ \left\{ u,v\right\} \mid u,v\in V\land u\neq v\right\}
\]
\end_inset
\end_layout
\begin_layout Notation*
\begin_inset Index idx
status open
\begin_layout Plain Layout
\begin_inset Formula $K_{n,m}$
\end_inset
\end_layout
\end_inset
\begin_inset Formula $K_{n,m}=\left(V,E\right)$
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bezeichnet den vollständig bipartiten Graph mit
\begin_inset Formula $V=V_{1}\varoplus V_{2}$
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,
\begin_inset Formula $\left|V_{1}\right|=n$
\end_inset
,
\begin_inset Formula $\left|V_{2}\right|=m$
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und
\begin_inset Formula
\[
E=\left\{ \left\{ v_{1},v_{2}\right\} \mid v_{1}\in V_{1}\land v_{2}\in V_{2}\right\}
\]
\end_inset
\end_layout
\begin_layout Definition
\begin_inset Index idx
status open
\begin_layout Plain Layout
planar
\end_layout
\end_inset
Ein Graph
\begin_inset Formula $G=\left(V,E\right)$
\end_inset
heißt
\noun on
planar,
\noun default
falls es eine Darstellung von
\begin_inset Formula $G$
\end_inset
in
\begin_inset Formula $\mathbb{R}^{2}$
\end_inset
gibt, bei der sich keine zwei Kanten kreuzen.
Das heißt, dass die Jordan-Kurven, die die Kanten darstellen, sich nur
in Endpunkten berühren.
\end_layout
\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open
\begin_layout Plain Layout
\align center
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{tikzpicture}
\end_layout
\begin_layout Plain Layout
\backslash
GraphInit[vstyle=Hasse]
\end_layout
\begin_layout Plain Layout
\backslash
SetGraphUnit{2}
\end_layout
\begin_layout Plain Layout
\backslash
Vertex[a=0,d=0]{1}
\end_layout
\begin_layout Plain Layout
\backslash
begin{scope}[rotate=90]
\end_layout
\begin_layout Plain Layout
\backslash
Vertices{circle}{2,3,4}
\end_layout
\begin_layout Plain Layout
\backslash
end{scope}
\end_layout
\begin_layout Plain Layout
\backslash
Edges(1,2,3,4,1)
\end_layout
\begin_layout Plain Layout
\backslash
Edge(1)(3)
\end_layout
\begin_layout Plain Layout
\backslash
Edge(2)(4)
\end_layout
\begin_layout Plain Layout
\backslash
end{tikzpicture}
\end_layout
\end_inset
\begin_inset space \hspace{}
\length 20mm
\end_inset
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{tikzpicture}
\end_layout
\begin_layout Plain Layout
\backslash
SetGraphUnit{2}
\end_layout
\begin_layout Plain Layout
\backslash
GraphInit[vstyle=Hasse]
\end_layout
\begin_layout Plain Layout
\backslash
Vertex{1}
\end_layout
\begin_layout Plain Layout
\backslash
WE(1){2}
\end_layout
\begin_layout Plain Layout
\backslash
EA(1){3}
\end_layout
\begin_layout Plain Layout
\backslash
NO(1){4}
\end_layout
\begin_layout Plain Layout
\backslash
SO(1){5}
\end_layout
\begin_layout Plain Layout
\backslash
AddVertexColor{scndcolor}{4,5}
\end_layout
\begin_layout Plain Layout
\backslash
Edge(1)(4)
\end_layout
\begin_layout Plain Layout
\backslash
Edge(2)(4)
\end_layout
\begin_layout Plain Layout
\backslash
Edge(3)(4)
\end_layout
\begin_layout Plain Layout
\backslash
Edge(1)(5)
\end_layout
\begin_layout Plain Layout
\backslash
Edge(2)(5)
\end_layout
\begin_layout Plain Layout
\backslash
Edge(3)(5)
\end_layout
\begin_layout Plain Layout
\backslash
end{tikzpicture}
\end_layout
\end_inset
\begin_inset Caption Standard
\begin_layout Plain Layout
\begin_inset Formula $K_{4}$
\end_inset
und
\begin_inset Formula $K_{2,3}$
\end_inset
sind planar.
\end_layout
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Proposition
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
thlabel{k5-nicht-planar}
\end_layout
\end_inset
\begin_inset Argument 1
status open
\begin_layout Plain Layout
\begin_inset Formula $K_{5}$
\end_inset
ist nicht planar|hide
\end_layout
\end_inset
\begin_inset Formula $K_{5}$
\end_inset
ist nicht planar.
\end_layout
\begin_layout Proof
Sei o.B.d.A
\begin_inset Formula $V\coloneqq\left\{ 1,2,3,4,5\right\} $
\end_inset
.
Bette die Knoten und alle zu
\begin_inset Formula $1$
\end_inset
inzidenten Kanten beliebig in die Ebene ein.
Dann existieren zwei Knoten (hier o.B.d.A.
\begin_inset Formula $2$
\end_inset
und
\begin_inset Formula $4$
\end_inset
), sodass der Kreis
\begin_inset Formula $C\coloneqq\left\{ \left\{ 4,1\right\} ,\left\{ 1,2\right\} ,\left\{ 2,4\right\} \right\} $
\end_inset
die Ebene in zwei Teile teilt:
\begin_inset Formula $\text{Äußeres}\left(C\right)$
\end_inset
und
\begin_inset Formula $\text{Inneres}\left(C\right)$
\end_inset
, wobei
\begin_inset Formula $3\in\text{Inneres}\left(C\right)$
\end_inset
und
\begin_inset Formula $5\in\text{Äußeres}\left(C\right)$
\end_inset
.
Jede Kante zwischen Innerem und Äußerem von
\begin_inset Formula $C$
\end_inset
muss den Rand von
\begin_inset Formula $C$
\end_inset
kreuzen.
Die Kante
\begin_inset Formula $\left\{ 3,5\right\} $
\end_inset
kann also nicht kreuzungsfrei gezogen werden (Jordan’scher Kurvensatz).
\begin_inset Float figure
wide false
sideways false
status open
\begin_layout Plain Layout
\align center
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{tikzpicture}
\end_layout
\begin_layout Plain Layout
\backslash
Vertex{1}
\end_layout
\begin_layout Plain Layout
\backslash
Vertex[a=45,d=3]{2}
\end_layout
\begin_layout Plain Layout
\backslash
Vertex[a=135,d=2]{5}
\end_layout
\begin_layout Plain Layout