
TL;DR
This survey introduces and explores various notions of activity in graphs and matroids, highlighting their role in expanding the Tutte polynomial and connecting to broader combinatorial and topological concepts.
Contribution
It provides a comprehensive overview of activity concepts for graphs and matroids, including structural theorems and connections to algebraic and topological combinatorics.
Findings
Survey of multiple activity notions for graphs and matroids
Descriptions of structural theorems related to activities
Connections to shellability and algebraic combinatorics
Abstract
Unlike Whitney's definition of the corank-nullity generating function , Tutte's definition of his now eponymous polynomial requires a total order on the edges of which the polynomial is a posteriori independent. Tutte presented his definition in terms of internal and external activities of maximal spanning forests. Although Tutte's original definition may appear somewhat ad hoc upon first inspection, subsequent work by various researchers has demonstrated that activity is a deep combinatorial concept. In this survey, we provide an introduction to activities for graphs and matroids. Our primary goal is to survey several notions of activity for graphs which admit expansions of the Tutte polynomial. Additionally, we describe some fundamental structural theorems, and outline connections to the topological notion of shellability as well as several topics in algebraic…
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Advanced Graph Theory Research · Topological and Geometric Data Analysis
