Perfect Discrete Morse Functions on Triangulated 3-Manifolds

Abstract

Abstract. This work is focused on characterizing the existence of a perfect discrete Morse function on a triangulated 3-manifold M , that is, a discrete Morse function satisfying that the numbers of critical simplices coincide with the corresponding Betti numbers. We reduce this problem to the existence of such kind of function on a spine L of M , that is, a 2-subcomplex L such that M − Δ collapses to L, where Δ is a tetrahedron of M . Also, considering the decomposition of every 3-manifold into prime factors, we prove that if every prime factor of M admits a perfect discrete Morse function, then M admits such kind of function.

Parameterized complexity of discrete morse theory

“…Joswig and Pfetsch [26] prove that if you can solve Erasability in the spine of a 2-simplicial complex in polynomial time, then you can solve Morse matching in the entire complex in polynomial time. The proof technique easily extends to 3manifolds, leading to the following lemma which has been mentioned in previous works [32,3]. In other words, answering Erasability on the spine is the only difficult part when solving Morse Matching on a 3manifold.…”

“…For general 2-complexes (and 3-manifolds), the problem reduces directly to finding a maximal alternating cycle-free matching on a spine, i.e., a bipartite graph representing the 1-and 2-cell adjacencies [3,26,32] (Lemma 1). To solve this problem, we propose an explicit algorithm for computing maximal alternating cycle-free matchings which is fixedparameter tractable in the treewidth of this bipartite graph (Theorem 5).…”

Analysis of Morse Matchings: Parameterized Complexity and Stable Matching

“…Using Poincaré's duality, the dual result from Lemma 3.5 follows. The following lemma, which has been mentioned in previous work [3,45], is a combination of Lemmata 3.5 and 3.7 for 3-manifolds together with the fact that χ(M ) = 0 for every 3-manifold M . In other words, if one can solve Erasability, then one can solve Morse Matching on a 3-manifold.…”

“…For general 2-complexes (and 3-manifolds), the problem reduces directly to finding a maximal alternating cycle-free matching on a spine, i.e. a bipartite graph representing the 1-and 2-cell adjacencies [3,38,43] (Lemma 3.8). To solve this problem, we propose an explicit algorithm for computing Morse matchings on bipartite graphs which is fixed-parameter tractable in the treewidth of the graph (Theorem 4.6).…”

“…Parameter: A non-negative integer k. Question: Is there a set of vertices D, |D| ≤ k, such that every vertex not in D is adjacent to at least one member of D? In addition there is also a parameterized reduction from Weighted Circuit Satisfiability of C [1,3] to Independent Set [20]. Therefore Independent Set is W[1]-complete, which means that if Independent Set can one day be shown to be in FPT, then W [1] = FPT.…”