-
Notifications
You must be signed in to change notification settings - Fork 3
/
Inoffizielles Skript.lyx
18721 lines (12696 loc) · 253 KB
/
Inoffizielles Skript.lyx
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
#LyX 2.2 created this file. For more info see http://www.lyx.org/
\lyxformat 508
\begin_document
\begin_header
\save_transient_properties true
\origin unavailable
\textclass scrbook
\begin_preamble
\usepackage{tikz}
\usepackage{tkz-graph}
\usetikzlibrary{calc}
\usetikzlibrary{decorations.pathreplacing}
\usepackage[tocfullflat]{tocstyle}
\deactivatetocstyle[toc]
\usetocstyle{standard}
\settocfeature{entryhook}{\large}
\settocfeature{entryvskip}{1\baselineskip plus .2pt}
\makeatletter
\newcommand\listpropositionname{\chapter*{Verzeichnis der Sätze}\vspace{-2em}}
\newcommand\listofpropositions{%
\section*{\listpropositionname}\@starttoc{lem}}
\makeatother
\renewcommand*{\VertexSmallMinSize}{20pt}
% All Vertices in Math Mode
\tikzset{VertexStyle/.append style={execute at begin node=$, execute at end node=$}}
\definecolor{scndcolor}{gray}{0.8}
\definecolor{thrdcolor}{gray}{0.4}
\usepackage{theoremref}
%\usepackage{breqn}
\usepackage{mdframed}
\usepackage{xstring}
\newenvironment{planargraphtheorem}{\begin{samepage}}{\end{samepage}}
% Formatierung der verschiedenen Theorem-Umgebungen
% Formatierung für Sätze
\newmdenv[
linewidth=2pt,
linecolor=black,
innertopmargin=-3pt,
skipabove=8mm,
skipbelow=6mm,
leftmargin=4mm,
hidealllines=true,
leftline=true,
nobreak=true
]{propstyle}
\let\oldprop\prop
\renewenvironment{prop}[1][]%
{%
\begin{planargraphtheorem}\begin{propstyle}
\IfEndWith{#1}{|hide}{\begin{oldprop}[]}{\begin{oldprop}[#1]}
\StrDel{#1}{|hide}[\listtitle]
\addcontentsline{lem}{section}{\listtitle}
} {%
\end{oldprop}
\end{propstyle}
\end{planargraphtheorem}
}%
% Formatierung für Lemmata
\newmdenv[
linewidth=2pt,
linecolor=gray,
innertopmargin=-3pt,
skipabove=4mm,
skipbelow=3mm,
leftmargin=4mm,
hidealllines=true,
leftline=true,
nobreak=true
]{lemstyle}
\let\oldlem\lem
\renewenvironment{lem}
{\begin{planargraphtheorem}\begin{lemstyle}\begin{oldlem}}
{\end{oldlem}\end{lemstyle}\end{planargraphtheorem}}
\catcode`\*=11
\let\oldlems\lem*
\catcode`\*=12
\renewenvironment{lem*}
{\begin{planargraphtheorem}\begin{lemstyle}\begin{oldlems}}
{\end{oldlems}\end{lemstyle}\end{planargraphtheorem}}
% Formatierung für Definitionen
\newmdenv[
linewidth=1pt,
linecolor=black,
innertopmargin=0mm,
innerbottommargin=3mm,
skipabove=5mm,
skipbelow=5mm,
hidealllines=true,
topline=true,
bottomline=true,
nobreak=true
]{defnstyle}
\LetLtxMacro\olddefn\defn
\renewenvironment{defn}
{\begin{planargraphtheorem}\begin{defnstyle}\begin{olddefn}}
{\end{olddefn}\end{defnstyle}\end{planargraphtheorem}}
\catcode`\*=11
\LetLtxMacro\olddefns\defn*
\catcode`\*=12
\renewenvironment{defn*}
{\begin{planargraphtheorem}\begin{defnstyle}\begin{olddefns}}
{\end{olddefns}\end{defnstyle}\end{planargraphtheorem}}
\LetLtxMacro\oldproblem\problem
\renewenvironment{problem}[1][]
{\begin{planargraphtheorem}\begin{oldproblem}[#1]\index{#1}}
{\end{oldproblem}\end{planargraphtheorem}}
% Abstände für Paragraphen und Unterpargraphen
\RedeclareSectionCommands[
beforeskip=-.7\baselineskip,
afterskip=1sp
]{paragraph}
\RedeclareSectionCommands[
beforeskip=.7\baselineskip,
afterskip=-1em
]{subparagraph}
% Abstände für Listen
\setlist[itemize]{
topsep=0.4\baselineskip,
parsep=0.1\baselineskip,
itemsep=0.1\baselineskip,
after=\vspace{.1\baselineskip}
}
\setlist[enumerate]{
topsep=0.4\baselineskip,
parsep=0.1\baselineskip,
itemsep=0.1\baselineskip,
after=\vspace{.1\baselineskip}
}
% Farben zum Einfärben von Graphen
\usepackage{xcolor}
\definecolor{color1}{named}{red}
\definecolor{color2}{named}{green}
\definecolor{color3}{named}{cyan}
\definecolor{color4}{named}{brown}
\definecolor{color5}{named}{yellow}
% Nummeriere Abbildugen nach Kapitel
\numberwithin{figure}{chapter}
% Formatierung für Abbildungen
\addtokomafont{captionlabel}{\small\sffamily}
\addtokomafont{caption}{\small}
\usepackage{caption}
\DeclareCaptionLabelSeparator{withvspace}{\\\vspace{.1\baselineskip}}
\captionsetup{
format=plain,
labelsep=withvspace,
justification=centerlast,
skip=\baselineskip,
position=below,
margin=7mm
}
% Vermeidung von Hurenkindern und Schusterjungen
\usepackage[all]{nowidow}
% QED-Symbol
\renewcommand{\qedsymbol}{\(\blacksquare\)}
\newcommand{\smallqed}{\renewcommand{\qedsymbol}{\(\square\)}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Color nodes with two colors.
% Code taken from
% http://tex.stackexchange.com/questions/55594/tikz-two-colored-circle-split
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\tikzset{circle split part fill/.style args={#1,#2}{%
alias=tmp@name, % Jake's idea !!
postaction={%
insert path={
\pgfextra{%
\pgfpointdiff{\pgfpointanchor{\pgf@node@name}{center}}%
{\pgfpointanchor{\pgf@node@name}{east}}%
\pgfmathsetmacro\insiderad{\pgf@x}
\fill[#1] (\pgf@node@name.base) ([xshift=-\pgflinewidth]\pgf@node@name.east) arc
(0:180:\insiderad-\pgflinewidth)--cycle;
\fill[#2] (\pgf@node@name.base) ([xshift=\pgflinewidth]\pgf@node@name.west) arc
(180:360:\insiderad-\pgflinewidth)--cycle;
}}}}}
% Geklauter Code für Gitter in tikz-Zeichnungen
\makeatletter
\def\grd@save@target#1{%
\def\grd@target{#1}}
\def\grd@save@start#1{%
\def\grd@start{#1}}
\tikzset{
grid with coordinates/.style={
to path={%
\pgfextra{%
\edef\grd@@target{(\tikztotarget)}%
\tikz@scan@one@point\grd@save@target\grd@@target\relax
\edef\grd@@start{(\tikztostart)}%
\tikz@scan@one@point\grd@save@start\grd@@start\relax
\draw[minor help lines] (\tikztostart) grid (\tikztotarget);
\draw[major help lines] (\tikztostart) grid (\tikztotarget);
\grd@start
\pgfmathsetmacro{\grd@xa}{\the\pgf@x/1cm}
\pgfmathsetmacro{\grd@ya}{\the\pgf@y/1cm}
\grd@target
\pgfmathsetmacro{\grd@xb}{\the\pgf@x/1cm}
\pgfmathsetmacro{\grd@yb}{\the\pgf@y/1cm}
\pgfmathsetmacro{\grd@xc}{\grd@xa + \pgfkeysvalueof{/tikz/grid with coordinates/major step}}
\pgfmathsetmacro{\grd@yc}{\grd@ya + \pgfkeysvalueof{/tikz/grid with coordinates/major step}}
\foreach \x in {\grd@xa,\grd@xc,...,\grd@xb}
\node[anchor=north] at (\x,\grd@ya) {\pgfmathprintnumber{\x}};
\foreach \y in {\grd@ya,\grd@yc,...,\grd@yb}
\node[anchor=east] at (\grd@xa,\y) {\pgfmathprintnumber{\y}};
}
}
},
minor help lines/.style={
help lines,
step=\pgfkeysvalueof{/tikz/grid with coordinates/minor step}
},
major help lines/.style={
help lines,
line width=\pgfkeysvalueof{/tikz/grid with coordinates/major line width},
step=\pgfkeysvalueof{/tikz/grid with coordinates/major step}
},
grid with coordinates/.cd,
minor step/.initial=.2,
major step/.initial=1,
major line width/.initial=2pt,
}
\makeatother
\makeatletter
\end_preamble
\use_default_options true
\begin_modules
eqs-within-sections
figs-within-sections
theorems-ams-bytype
theorems-ams-extended-bytype
algorithm2e
enumitem
\end_modules
\maintain_unincluded_children false
\language ngerman
\language_package default
\inputencoding auto
\fontencoding global
\font_roman "lmodern" "default"
\font_sans "lmss" "default"
\font_typewriter "lmtt" "default"
\font_math "auto" "auto"
\font_default_family default
\use_non_tex_fonts false
\font_sc false
\font_osf false
\font_sf_scale 100 100
\font_tt_scale 100 100
\graphics default
\default_output_format default
\output_sync 0
\bibtex_command default
\index_command default
\float_placement tbph
\paperfontsize 10
\spacing single
\use_hyperref true
\pdf_title "Mitschrieb Algorithmen für planare Graphen"
\pdf_author "Joshua Gleitze"
\pdf_bookmarks true
\pdf_bookmarksnumbered false
\pdf_bookmarksopen true
\pdf_bookmarksopenlevel 1
\pdf_breaklinks true
\pdf_pdfborder true
\pdf_colorlinks false
\pdf_backref false
\pdf_pdfusetitle true
\papersize default
\use_geometry false
\use_package amsmath 2
\use_package amssymb 1
\use_package cancel 1
\use_package esint 1
\use_package mathdots 1
\use_package mathtools 1
\use_package mhchem 1
\use_package stackrel 1
\use_package stmaryrd 2
\use_package undertilde 1
\cite_engine basic
\cite_engine_type default
\biblio_style plain
\use_bibtopic false
\use_indices false
\paperorientation portrait
\suppress_date true
\justification true
\use_refstyle 1
\index Index
\shortcut idx
\color #008000
\end_index
\secnumdepth 3
\tocdepth 3
\paragraph_separation indent
\paragraph_indentation default
\quotes_language german
\papercolumns 1
\papersides 1
\paperpagestyle default
\bullet 0 0 0 -1
\tracking_changes false
\output_changes false
\html_math_output 0
\html_css_as_file 0
\html_be_strict false
\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
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
listofpropositions
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
clearpage
\backslash
pagenumbering{arabic}
\end_layout
\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 $K_{n,m}=\left(V,E\right)$
\end_inset
bezeichnet den vollständig bipartiten Graph mit
\begin_inset Formula $V=V_{1}\varoplus V_{2}$
\end_inset
,
\begin_inset Formula $\left|V_{1}\right|=n$
\end_inset
,
\begin_inset Formula $\left|V_{2}\right|=m$
\end_inset
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