1% ============================================================================2% UQO Working Paper 32 — Sharp Zero-Forcing Bounds for Generalized Petersen Graphs3% Retypeset from AutoScholar Working Paper AS-WP-2026-0001 (v1, 2026-10-30)4% ============================================================================5\documentclass[12pt,letterpaper]{article}67% --- Encoding & Language ---8\usepackage[utf8]{inputenc}9\usepackage[T1]{fontenc}10\usepackage[english]{babel}1112% --- Page Layout ---13\usepackage[letterpaper,top=1in,bottom=1in,left=1in,right=1in]{geometry}1415% --- Typography ---16\usepackage{mathptmx}17\usepackage{microtype}18\usepackage{setspace}19\onehalfspacing2021% --- Math ---22\usepackage{amsmath,amssymb,amsthm}2324% --- Tables ---25\usepackage{booktabs}26\usepackage{tabularx}27\usepackage{multirow}28\usepackage{threeparttable}29\usepackage{array,longtable}30\usepackage{siunitx}31\sisetup{input-symbols = {()}, table-format = -1.3, parse-numbers = 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---75\theoremstyle{plain}76\newtheorem{theorem}{Theorem}[section]77\newtheorem{proposition}[theorem]{Proposition}78\newtheorem{lemma}[theorem]{Lemma}79\newtheorem{corollary}[theorem]{Corollary}80\newtheorem{conjecture}[theorem]{Conjecture}81\theoremstyle{definition}82\newtheorem{definition}[theorem]{Definition}83\newtheorem{assumption}[theorem]{Assumption}84\newtheorem{example}[theorem]{Example}85\newtheorem{observation}[theorem]{Observation}86\newtheorem{compresult}[theorem]{Computational Result}87\newtheorem{question}[theorem]{Question}88\theoremstyle{remark}89\newtheorem{remark}[theorem]{Remark}9091% --- Paper macros ---92\newcommand{\Z}{\operatorname{Z}}93\newcommand{\cl}{\operatorname{cl}}94\newcommand{\Aut}{\operatorname{Aut}}95\newcommand{\PP}[2]{P(#1,#2)}96\newcommand{\bd}{\partial}97\newcommand{\Fcnf}{\mathcal F}98\newcommand{\Gcnf}{\mathcal G}99\newcommand{\Nmax}{N_{\max}}100\newcommand{\cmax}{c_{\max}}101\newcommand{\code}[1]{\texttt{#1}}102103% ============================================================================104% METADATA105% ============================================================================106\newcommand{\WPnumber}{32}107\newcommand{\WPtitle}{Sharp Zero-Forcing Bounds for Generalized Petersen Graphs}108\newcommand{\WPsubtitle}{Computational Discovery and Certified Proofs for $P(n,5)$}109\newcommand{\WPdate}{October 2026}110\newcommand{\WPversion}{1.0}111\newcommand{\WPabstract}{\input{sections/abstract}}112\newcommand{\WPkeywords}{zero forcing, generalized Petersen graphs, minimum rank, forts, SAT certificates, DRAT, computer-assisted proof}113\newcommand{\WPjel}{C61, C63 (MSC 2020: 05C50, 05C69, 05C85, 68V15)}114115% --- Author ---116\newcommand{\WPauthor}{Simon-Pierre Boucher}117\newcommand{\WPaffiliation}{%118 Département des sciences administratives\\119 Université du Québec en Outaouais%120}121\newcommand{\WPemail}{simon-pierre.boucher@uqo.ca}122\newcommand{\WPaddress}{%123 Gatineau -- Pavillon Alexandre-Taché\\124 283, boulevard Alexandre-Taché\\125 Gatineau, Québec, Canada J9A 1L8%126}127128% ============================================================================129% DOCUMENT130% ============================================================================131\begin{document}132133\input{sections/titlepage}134135\input{sections/introduction}136\input{sections/related}137\input{sections/preliminaries}138\input{sections/upper}139\input{sections/lower}140\input{sections/framework}141\input{sections/results}142\input{sections/robustness}143\input{sections/discussion}144\input{sections/limitations}145\input{sections/conclusion}146147\newpage148\bibliography{references}149150\input{appendix/appendix}151152\end{document}153