Treewidth: A Key Parameter in Graph Theory and Its Applications
2025-01-14
This article delves into treewidth, a crucial parameter in graph theory. Defined using tree decompositions, treewidth characterizes graph structure and is closely related to algorithmic complexity. The article explores multiple equivalent definitions of treewidth, its structural properties, and computational methods. It then details its broad applications in sparse numerical linear algebra, Bayesian inference, game theory, low-dimensional topology, network science, and algebraic geometry. The author also discusses advances in related width parameters and how treewidth can improve the efficiency of graph algorithms.