Courcelle's theorem for triangulations

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 dim...

Full description

Saved in:

Bibliographic Details
Published in:	Journal of combinatorial theory. Series A Vol. 146; pp. 264 - 294
Main Authors:	Burton, Benjamin A., Downey, Rodney G.
Format:	Journal Article
Language:	English
Published:	Elsevier Inc 01.02.2017
Subjects:	3-Manifolds Discrete Morse theory Parameterised complexity Triangulations Turaev–Viro invariants 3-Manifolds Turaev–Viro invariants Discrete Morse theory Triangulations Parameterised complexity
ISSN:	0097-3165, 1096-0899
Online Access:	Get full text
Tags:	Add Tag No Tags, Be the first to tag this record!

Description
Summary:	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.
ISSN:	0097-3165 1096-0899
DOI:	10.1016/j.jcta.2016.10.001