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dual_decomposition: reduced-dual definition, verification, and step figures

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Add Definition 2.1 (reduced dual) and a remark on cubicity/planarity, plus an
experiment verifying it on the icosahedron/dodecahedron and four figures, one
per construction step.

reduced_dual.py builds G' = dodecahedron (dual of the icosahedron), applies the
construction, and confirms the result is a cubic, planar, simple graph whose
dual is a simple triangulation. Finding: the construction is an n -> n-2
reduction (12 -> 10 here), not n-1, since the single apex v_n collapses one more
vertex than a standard pentagon re-triangulation; the result also re-introduces
degree-3 and degree-4 vertices (degree seq [7,5,5,5,5,5,5,4,4,3]).

draw_reduced_dual_steps.py renders fig_reduced_dual_step1..4.png, embedded as a
2x2 grid after the definition.

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
This commit is contained in:
didericis
2026-05-22 18:50:38 -04:00
parent bd8526eb11
commit 1791b68f4a
12 changed files with 508 additions and 27 deletions
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papers/dual_decomposition_minimal_counterexamples/paper.fls
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@@ -227,10 +227,32 @@ INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm1
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm7.tfm
OUTPUT paper.pdf
INPUT /usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map
INPUT ./fig_reduced_dual_step1.png
INPUT ./fig_reduced_dual_step1.png
INPUT fig_reduced_dual_step1.png
INPUT ./fig_reduced_dual_step1.png
INPUT ./fig_reduced_dual_step1.png
INPUT ./fig_reduced_dual_step2.png
INPUT ./fig_reduced_dual_step2.png
INPUT fig_reduced_dual_step2.png
INPUT ./fig_reduced_dual_step2.png
INPUT ./fig_reduced_dual_step2.png
INPUT ./fig_reduced_dual_step3.png
INPUT ./fig_reduced_dual_step3.png
INPUT fig_reduced_dual_step3.png
INPUT ./fig_reduced_dual_step3.png
INPUT ./fig_reduced_dual_step3.png
INPUT ./fig_reduced_dual_step4.png
INPUT ./fig_reduced_dual_step4.png
INPUT fig_reduced_dual_step4.png
INPUT ./fig_reduced_dual_step4.png
INPUT ./fig_reduced_dual_step4.png
INPUT paper.aux
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