dual_decomposition: reduced-dual definition, verification, and step figures
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>
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[]\OT1/cmr/m/n/10 List the five degree-$2$ ver-tices in clock-wise or-der aroun
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d $\OML/cmm/m/it/10 F$ \OT1/cmr/m/n/10 as $\OML/cmm/m/it/10 A \OT1/cmr/m/n/10 =
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(\OML/cmm/m/it/10 A[]; A[]; A[]; A[]; A[]\OT1/cmr/m/n/10 )$.
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[]
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<fig_reduced_dual_step1.png, id=17, 517.79329pt x 499.08812pt>
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File: fig_reduced_dual_step1.png Graphic file (type png)
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File: fig_reduced_dual_step4.png Graphic file (type png)
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[2] [3 <./fig_reduced_dual_step1.png> <./fig_reduced_dual_step2.png> <./fig_red
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uced_dual_step3.png> <./fig_reduced_dual_step4.png>] (./paper.aux) )
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