Robertson–Seymour theorem

Every minor-closed family of finite graphs can be characterized by a finite set of forbidden minors. The theorem follows from the well-quasi-ordering of finite graphs under the graph-minor relation.

Graph minor: A graph obtained from another by deleting vertices or edges and contracting edges. The theorem’s ordering relation compares graphs by whether one is a minor of another.
Wagner's theorem: A characterization of planar graphs as graphs with neither K5 nor K3,3 as a minor. It is a classical finite forbidden-minor characterization of a minor-closed family.
Tree decomposition: A representation of a graph as overlapping vertex sets arranged in a tree, subject to connectivity conditions. Tree-like structure is a key tool for organizing graphs that avoid a fixed minor.
Neil Robertson: A British mathematician known for foundational work in graph theory, especially graph minors. He co-developed the graph-minor theory with Paul Seymour.
Minor-closed graph family: A class of graphs that contains every minor of each graph it contains. The theorem applies precisely to classes closed under taking graph minors.
Planar graph: A graph that can be drawn in the plane without edges crossing except at shared endpoints. Planarity exemplifies a property captured by finitely many forbidden minors.
Treewidth: A measure of how closely a graph resembles a tree, defined through its tree decompositions. Bounded treewidth supplies tractable structure in important stages of minor theory.
Paul Seymour: A British mathematician whose work in graph theory includes the theory of graph minors. He co-developed the theorem and its extensive structural framework with Neil Robertson.
Well-quasi-ordering: A quasi-order in which every infinite sequence contains an earlier element below a later one. The graph-minor relation’s well-quasi-ordering forces each class’s minimal excluded graphs to be finite.
Graph minor theorem: The theorem that finite graphs are well-quasi-ordered by the graph-minor relation. This is the central structural result from which the finite-obstruction conclusion follows.

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