Prove outerplanarity and draft edge-flip resolution algorithm
- Promote Prop 3.1 (outerplanarity of level subgraphs) to Theorem 3.1 with a proof by contradiction via a BFS-path argument; drop the $n \leq 10$ caveat and the now-resolved open question. - Add Section 5 "An edge-flip resolution algorithm": apex classification of $L_k$-edges, bridge lemma, cross-level flip pass, definition of tricky-everywhere odd cycles and facial depth (seeded from inner faces with $\geq 2$ outer-face edges), and the depth-guided flip procedure. Observation 5.5 records empirical termination at $n = 9, 10, 11$; Question 5.6 asks if it holds in general. - Add experiments/depth_monovariant_check.py (sanity check over triangulation iso-classes, confirms the count-of-tricky-faces monovariant strictly decreases per flip on all 1400 tricky configs at $n \leq 11$), viz_cycling.py and debug_cycling.py, and cycling_visualization.png illustrating the depth-definition fix. Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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|
||||
Package hyperref Warning: Token not allowed in a PDF string (Unicode):
|
||||
(hyperref) removing `math shift' on input line 370.
|
||||
|
||||
[5] [6] [7] (./paper.aux)
|
||||
Package rerunfilecheck Info: File `paper.out' has not changed.
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||||
(rerunfilecheck) Checksum: EEEB5CE29153D98296A2AD7EF6DBBA0A;2386.
|
||||
(rerunfilecheck) Checksum: C4EC0643C07D7F8B157295F889BACC80;3432.
|
||||
)
|
||||
Here is how much of TeX's memory you used:
|
||||
8910 strings out of 478268
|
||||
137894 string characters out of 5846347
|
||||
439445 words of memory out of 5000000
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||||
26840 multiletter control sequences out of 15000+600000
|
||||
8935 strings out of 478268
|
||||
137960 string characters out of 5846347
|
||||
440651 words of memory out of 5000000
|
||||
26851 multiletter control sequences out of 15000+600000
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||||
475834 words of font info for 54 fonts, out of 8000000 for 9000
|
||||
1302 hyphenation exceptions out of 8191
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||||
69i,9n,76p,482b,426s stack positions out of 10000i,1000n,20000p,200000b,200000s
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||||
</usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmcsc10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi6.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy6.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmtt10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb>
|
||||
Output written on paper.pdf (5 pages, 235833 bytes).
|
||||
69i,9n,76p,396b,463s stack positions out of 10000i,1000n,20000p,200000b,200000s
|
||||
</usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx10.pfb
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||||
></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmcsc10.pfb
|
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></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb>
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</usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi6.pfb></
|
||||
usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb></us
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||||
r/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi8.pfb></usr/
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||||
local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb></usr/lo
|
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cal/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb></usr/local
|
||||
/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr8.pfb></usr/local/te
|
||||
xlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb></usr/local/tex
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||||
live/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy5.pfb></usr/local/texli
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||||
ve/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy6.pfb></usr/local/texlive
|
||||
/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb></usr/local/texlive/2
|
||||
022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb></usr/local/texlive/20
|
||||
22/texmf-dist/fonts/type1/public/amsfonts/cm/cmti8.pfb></usr/local/texlive/2022
|
||||
/texmf-dist/fonts/type1/public/amsfonts/cm/cmtt10.pfb></usr/local/texlive/2022/
|
||||
texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb></usr/local/texlive/2
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||||
022/texmf-dist/fonts/type1/public/amsfonts/symbols/msbm10.pfb>
|
||||
Output written on paper.pdf (7 pages, 254719 bytes).
|
||||
PDF statistics:
|
||||
212 PDF objects out of 1000 (max. 8388607)
|
||||
169 compressed objects within 2 object streams
|
||||
40 named destinations out of 1000 (max. 500000)
|
||||
89 words of extra memory for PDF output out of 10000 (max. 10000000)
|
||||
264 PDF objects out of 1000 (max. 8388607)
|
||||
216 compressed objects within 3 object streams
|
||||
55 named destinations out of 1000 (max. 500000)
|
||||
129 words of extra memory for PDF output out of 10000 (max. 10000000)
|
||||
|
||||
|
||||
Reference in New Issue
Block a user