% ============================================================================
% UQO Working Paper 32 — Zero-Forcing Bounds for Generalized Petersen Graphs
% Retypeset from AutoScholar Working Paper AS-WP-2026-0001 (v1, 2026-10-30)
% ============================================================================
\documentclass[12pt,letterpaper]{article}

% --- Encoding & Language ---
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[english]{babel}

% --- Page Layout ---
\usepackage[letterpaper,top=1in,bottom=1in,left=1in,right=1in]{geometry}

% --- Typography ---
\usepackage{mathptmx}
\usepackage{microtype}
\usepackage{setspace}
\onehalfspacing

% --- Math ---
\usepackage{amsmath,amssymb,amsthm}

% --- Tables ---
\usepackage{booktabs}
\usepackage{tabularx}
\usepackage{multirow}
\usepackage{threeparttable}
\usepackage{array,longtable}
\usepackage{siunitx}
\sisetup{input-symbols = {()}, table-format = -1.3, parse-numbers = false}
\newcolumntype{L}[1]{>{\raggedright\arraybackslash}p{#1}}
\newcolumntype{R}[1]{>{\raggedleft\arraybackslash}p{#1}}

% --- Figures ---
\usepackage{graphicx}
\usepackage{subcaption}
\usepackage{float}
\graphicspath{{figures/}}

% --- Colors & Links ---
\usepackage[dvipsnames]{xcolor}
\usepackage[colorlinks=true,linkcolor=NavyBlue,citecolor=NavyBlue,urlcolor=NavyBlue,breaklinks=true]{hyperref}
\hypersetup{pdftitle={Zero-Forcing Bounds for Generalized Petersen Graphs: Computational Discovery and Certified Proofs for P(n,5)},pdfauthor={Simon-Pierre Boucher}}
\makeatletter\g@addto@macro{\UrlBreaks}{\UrlOrds}\makeatother % line breaks inside file paths

% --- Bibliography ---
\usepackage[authoryear,round,semicolon]{natbib}
\bibliographystyle{aer}

% --- Captions ---
\usepackage[font=small,labelfont=bf,labelsep=period,justification=justified,singlelinecheck=false]{caption}

% --- Headers & Footers ---
\usepackage{fancyhdr}
\pagestyle{fancy}
\fancyhf{}
\renewcommand{\headrulewidth}{0pt}
\fancyfoot[C]{\thepage}

% --- Section Formatting ---
\usepackage{titlesec}
\titleformat{\section}{\large\bfseries}{\thesection.}{0.5em}{}
\titleformat{\subsection}{\normalsize\bfseries}{\thesubsection.}{0.5em}{}
\titleformat{\subsubsection}{\normalsize\itshape}{\thesubsubsection.}{0.5em}{}

% --- Appendix Support ---
\usepackage[toc,page]{appendix}

% --- Misc ---
\usepackage{enumitem}
\setlist{nosep,leftmargin=*}

% --- Theorem environments (shared counter, numbered within section, as in the original) ---
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{conjecture}[theorem]{Conjecture}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{assumption}[theorem]{Assumption}
\newtheorem{example}[theorem]{Example}
\newtheorem{observation}[theorem]{Observation}
\newtheorem{compresult}[theorem]{Computational Result}
\newtheorem{question}[theorem]{Question}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}

% --- Paper macros ---
\newcommand{\Z}{\operatorname{Z}}
\newcommand{\cl}{\operatorname{cl}}
\newcommand{\Aut}{\operatorname{Aut}}
\newcommand{\PP}[2]{P(#1,#2)}
\newcommand{\bd}{\partial}
\newcommand{\Fcnf}{\mathcal F}
\newcommand{\Gcnf}{\mathcal G}
\newcommand{\Nmax}{N_{\max}}
\newcommand{\cmax}{c_{\max}}
\newcommand{\code}[1]{\texttt{#1}}

% ============================================================================
% METADATA
% ============================================================================
\newcommand{\WPnumber}{32}
\newcommand{\WPtitle}{Zero-Forcing Bounds for Generalized Petersen Graphs}
\newcommand{\WPsubtitle}{Computational Discovery and Certified Proofs for $P(n,5)$}
\newcommand{\WPdate}{October 2026}
\newcommand{\WPversion}{1.0}
\newcommand{\WPabstract}{\input{sections/abstract}}
\newcommand{\WPkeywords}{zero forcing, generalized Petersen graphs, minimum rank, forts, SAT certificates, DRAT, computer-assisted proof}
\newcommand{\WPjel}{C61, C63 (MSC 2020: 05C50, 05C69, 05C85, 68V15)}

% --- Author ---
\newcommand{\WPauthor}{Simon-Pierre Boucher}
\newcommand{\WPaffiliation}{%
  Département des sciences administratives\\
  Université du Québec en Outaouais%
}
\newcommand{\WPemail}{simon-pierre.boucher@uqo.ca}
\newcommand{\WPaddress}{%
  Gatineau -- Pavillon Alexandre-Taché\\
  283, boulevard Alexandre-Taché\\
  Gatineau, Québec, Canada J9A 1L8%
}

% ============================================================================
% DOCUMENT
% ============================================================================
\begin{document}

\input{sections/titlepage}

\input{sections/introduction}
\input{sections/related}
\input{sections/preliminaries}
\input{sections/upper}
\input{sections/lower}
\input{sections/framework}
\input{sections/results}
\input{sections/robustness}
\input{sections/discussion}
\input{sections/limitations}
\input{sections/conclusion}

\newpage
\bibliography{references}

\input{appendix/appendix}

\end{document}