The monadic second-order logic of graphs. I. recognizable sets of finite graphs
Information and Computation
A linear time algorithm for finding tree-decompositions of small treewidth
STOC '93 Proceedings of the twenty-fifth annual ACM symposium on Theory of computing
A computationally intractable problem on simplicial complexes
Computational Geometry: Theory and Applications
Efficient and constructive algorithms for the pathwidth and treewidth of graphs
Journal of Algorithms
Grow & fold: compression of tetrahedral meshes
Proceedings of the fifth ACM symposium on Solid modeling and applications
Efficiency of a Good But Not Linear Set Union Algorithm
Journal of the ACM (JACM)
On discrete Morse functions and combinatorial decompositions
Discrete Mathematics
The Parameterized Complexity of Some Problems in Logic and Linguistics
LFCS '94 Proceedings of the Third International Symposium on Logical Foundations of Computer Science
An Improved Algorithm for Finding Tree Decompositions of Small Width
WG '99 Proceedings of the 25th International Workshop on Graph-Theoretic Concepts in Computer Science
Optimal discrete Morse functions for 2-manifolds
Computational Geometry: Theory and Applications
Applications of Forman's Discrete Morse Theory to Topology Visualization and Mesh Compression
IEEE Transactions on Visualization and Computer Graphics
Computing Optimal Morse Matchings
SIAM Journal on Discrete Mathematics
Combinatorial construction of morse-smale complexes for data analysis and visualization
Combinatorial construction of morse-smale complexes for data analysis and visualization
Determining whether a simplicial 3-complex collapses to a 1-complex is NP-complete
DGCI'08 Proceedings of the 14th IAPR international conference on Discrete geometry for computer imagery
simpcomp: a GAP toolbox for simplicial complexes
ACM Communications in Computer Algebra
Detecting genus in vertex links for the fast enumeration of 3-manifold triangulations
Proceedings of the 36th international symposium on Symbolic and algebraic computation
Theory and Algorithms for Constructing Discrete Morse Complexes from Grayscale Digital Images
IEEE Transactions on Pattern Analysis and Machine Intelligence
SOFSEM'05 Proceedings of the 31st international conference on Theory and Practice of Computer Science
Perfect discrete morse functions on triangulated 3-manifolds
CTIC'12 Proceedings of the 4th international conference on Computational Topology in Image Context
Parameterized Complexity
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Optimal Morse matchings reveal essential structures of cell complexes which lead to powerful tools to study discrete geometrical objects, in particular discrete 3-manifolds. However, such matchings are known to be NP-hard to compute on 3-manifolds, through a reduction to the erasability problem. Here, we refine the study of the complexity of problems related to discrete Morse theory in terms of parameterized complexity. On the one hand we prove that the erasability problem is W[P]-complete on the natural parameter. On the other hand we propose an algorithm for computing optimal Morse matchings on triangulations of 3-manifolds which is fixed-parameter tractable in the treewidth of the bipartite graph representing the adjacency of the 1- and 2-simplexes. This algorithm also shows fixed parameter tractability for problems such as erasability and maximum alternating cycle-free matching. We further show that these results are also true when the treewidth of the dual graph of the triangulated 3-manifold is bounded. Finally, we investigate the respective treewidths of simplicial and generalized triangulations of 3-manifolds.