A note on the pure Morse complex of a graph

Abstract

The goal of this work is to study the structure of the pure Morse complex of a graph, that is, the simplicial complex given by the set of all possible classes of discrete Morse functions (in Forman's sense) defined on it. First, we characterize the pure Morse complex of a tree and prove that it is collapsible. In order to study the general case, we consider all the spanning trees included in a given graph G and we express the pure Morse complex of G as the union of all pure Morse complexes corresponding to such trees.

A Simplicial Complex is Uniquely Determined by Its Set of Discrete Morse Functions

“…A discrete Morse funcion f on an abstract simplicial complex K is a map f : K → R satisfying, for every σ ∈ K, (1) |{τ σ | f (τ ) ≤ f (σ)}| ≤ 1 and (2) |{ν ≺ σ | f (ν) ≥ f (σ)}| ≤ 1. Here |A| denotes the cardinality of the set A.…”

“…This proves that q ∈ G(C 2 ). Now, since Q has a complete 1-skeleton then we can form a cycle in Q (1) containing all the vertices of Q. The corresponding f -cycle of index 0 has as a preimage by F an f -cycle of index k − 1 with n + 2 regular pairs.…”

“…This gives a well-defined isomorphism F | M(K (1) ) : M(K (1) ) → M(L (1) ). By Theorem 3.5 there exists an isomorphism f : K (1) → L (1) with f (v) = s(F (v, e)) for any e v. Note that for every edge xy ∈ K we have F (x, xy) = (f (x), f (x)f (y)). We will show by induction that for any (n + 1)-simplex v 0 · · · v n+1 , F (v 0 · · · v n , v 0 · · · v n v n+1 ) = (f (v 0 ) · · · f (v n ), f (v 0 ) · · · f (v n )f (v n+1 )).…”

Shellability of complexes of directed trees

“…The subcomplex of a complex of directed trees generated by its maximal facets was studied in [1] and [5].…”

Discrete Morse functions, vector fields, and homological sequences on trees

“…Inspired by Nicolaescu [1], R. Ayala et al [2] studied the homological sequence of a discrete Morse function by introducing the notion of f and g being homologically equivalent, and they counted the number of excellent discrete Morse functions on all graphs [6]. In addition, the same authors counted the number of discrete Morse functions on a graph up to Forman equivalence in [4]. In this paper, we desire to combine these two questions by investigating the combination of the two notions of equivalence.…”