Under Review Elsevier · Discrete Applied Mathematics · Special Issue LAGOS 2025
Unit-Region Factorization for Empty-Region Proximity Graphs
Summary
A fixed-template empty-region rule assigns a Borel region \(S(p,q)\) to a candidate pair and retains the edge when that region contains no other site. The manuscript isolates the unit-volume constant \(a_K\) for similarity-copy regions and shows how the same scalar controls exact Poisson void probabilities, incident-edge length intensities, mean out-degree, and normalized incident-edge length laws.
$$ \lambda_d\!\left(S(p,q)\right)=\ell_{p,q}^{d}a_K,\qquad \mathbb{P}\!\left\{\Phi\!\left(S(p,q)\right)=0\right\}=\exp\!\left\{-\rho a_K\ell_{p,q}^{d}\right\} $$
Context
The manuscript is framed as a local scalar reduction: once a candidate-pair region is a translated, rotated, and uniformly scaled copy of K, the stochastic calculations depend on geometry through a single finite-volume constant.
Main contributions
- Formalizes pairwise empty-region rules through a normalized Borel template K.
- Derives the volume factorization \(\lambda_d(S(p,q))=\ell_{p,q}^{d}a_K\).
- Transfers the local scalar a_K into homogeneous Poisson and Palm calculations.
- Catalogs constants for Gabriel, relative-neighborhood, beta-skeleton, Veltkamp-type, finite-template, and stepping-stone regions.