Abstract
This chapter discusses an intensive theorem by the combinatorial logician Martin Grohe, building on Seymour’s pathbreaking work with Neil Robertson on the complexity of properties defined by excluding graph minors. No one has ever provided a logic that captures polynomial-time decidability on graphs without referencing an ordering of the vertices; Grohe’s theorem does so with restriction to minor-excluding graphs. One deep consequence is that the notoriously open Graph Isomorphism problem becomes polynomial-time decidable for graphs that obey the “promise” of excluding any fixed graph H as a minor.
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