Highly unbreakable graph with a fixed excluded minor are almost rigid
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A set X ⊆ V(G) in a graph G is (q,k)-unbreakable if every separation (A,B) of order at most k in G satisfies |A ∩ X| ≤ q or |B ∩ X| ≤ q. In this paper, we prove the following result: If a graph G excludes a fixed complete graph K_h as a minor and satisfies certain unbreakability guarantees, then G is almost rigid in the following sense: the vertices of G can be partitioned in an isomorphism-invariant way into a part inducing a graph of bounded treewidth and a part that admits a small isomorphism-invariant family of labelings. This result is the key ingredient in the fixed-parameter algorithm for Graph Isomorphism parameterized by the Hadwiger number of the graph, which is presented in a companion paper.
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