Global Structure of Minor-Free Graphs and Beyond

A graph consists of a set of vertices, representing objects, and edges between pairs of vertices, representing connections between these objects. Graphs are used to model networks, such as transportation networks and social networks, along with other complex problems such as disease transmission and molecular interaction. A graph minor identifies a substructure within a graph, so a minor-free graph is a graph that does not have a particular substructure. This thesis explores the global structure of minor-free graphs and graphs that resemble minor-free graphs, such as graphs that can be transformed into minor-free graphs while roughly preserving locality.