Courcelle's theorem for triangulations

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.