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Courcelle's theorem for triangulations (1403.2926v1)

Published 12 Mar 2014 in math.GT, cs.CC, cs.CG, and math.CO

Abstract: In graph theory, Courcelle's theorem essentially states that, if an algorithmic problem can be formulated in monadic second-order logic, then it can be solved in linear time for graphs of bounded treewidth. We prove such a metatheorem for a general class of triangulations of arbitrary fixed dimension d, including all triangulated d-manifolds: if an algorithmic problem can be expressed in monadic second-order logic, then it can be solved in linear time for triangulations whose dual graphs have bounded treewidth. We apply our results to 3-manifold topology, a setting with many difficult computational problems but very few parameterised complexity results, and where treewidth has practical relevance as a parameter. Using our metatheorem, we recover and generalise earlier fixed-parameter tractability results on taut angle structures and discrete Morse theory respectively, and prove a new fixed-parameter tractability result for computing the powerful but complex Turaev-Viro invariants on 3-manifolds.

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