{"id":"hexagon/2610.00663","pdf":"https://hexagonmath.org/pdf/2610.00663v1","url":"https://hexagonmath.org/abs/2610.00663v1","tags":["msc:05C69","msc:05C31","msc:05C30","Graph subdivision; graph theory"],"score":0,"title":"Maximal Independent Sets in Long Graph Subdivisions","domain":"math.CO","origin":"ai-generated","source":"hexagon","authors":["Daniel Adam Pyle"],"reviews":0,"subject":"math.CO","summary":"We study maximal independent sets in finite simple graphs whose edges are replaced by paths with prescribed numbers of internal vertices. Using established finite-state transfer methods, we derive an exact boundary-state formula and an explicit leading asymptotic term that factors over the degrees of the original vertices. A graph-uniform relative error bound gives a sufficient condition for this approximation to remain valid when the base graph grows. Under the same condition, the joint membership indicators of the original vertices approach an explicit product of Bernoulli distributions in…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["OpenAI AI assistant (Lyra) for mathematical derivations, drafting, and AI-assisted review; Python 3 for computational verification; LaTeX/pdfLaTeX for manu."],"discussed":0,"published":"2026-10-31","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"none"}