Fundamental Groupoids for Graphs

TL;DR

This paper introduces a new fundamental groupoid concept for graphs, establishing functorial relationships, properties of graph products, and a van Kampen theorem, advancing the algebraic understanding of graph structures.

Contribution

It develops a $ imes$-homotopy fundamental groupoid for graphs, linking it to graph categories and generalizing previous fundamental group concepts.

Findings

Established a functorial relationship to the 2-category of graphs

Developed a groupoid product respecting graph products

Proved a van Kampen Theorem for these groupoids

Abstract

In this paper, we develop a $\times$-homotopy fundamental groupoid for graphs, and show a functorial relationship to the 2-category of graphs. We further explore the fundamental groupoid of graph products and develop a groupoid product which respects the graph product. A van Kampen Theorem for these groupoids is provided. Finally, we generalize previous work on a fundamental group for graphs, developing a looped walk groupoid and showing a connection to the polyhedral complex of graph morphisms.