Gradient Vector Fields of Discrete Morse Functions are Minimum Spanning Forests

Nicolas Boutry (1) , Laurent Najman (2)

1 LRDE - Laboratoire de Recherche et de Développement de l'EPITA
2 LIGM - Laboratoire d'Informatique Gaspard-Monge

In this paper, we prove that discrete Morse functions are equivalent to simplicial stacks under reasonable constraints. We also show that, as in Discrete Morse Theory, we can see the GVF of a simplicial stack (seen as a discrete Morse function) as the only relevant information we should consider. Last, but not least, we prove that the Minimum Spanning Forest on the dual graph of a simplicial stack (or a discrete Morse function) is equal to the GVF of the initial function. In other words, the GVF of a discrete Morse function is related to a classic combinatorial minimization problem. This paper is the sequel of a sequence of papers showing that strong relations exist between different domains: Topology, Discrete Morse Theory, Topological Data Analysis and Mathematical Morphology.

Mots clés: Topological Data Analysis; Mathematical Morphology; Discrete Morse Theory; Simplicial Stacks; Minimum Spanning Forest

Domaines: Topologie algébrique [math.AT]; Topologie géométrique [math.GT]; Mathématique discrète [cs.DM]

Nicolas Boutry, Laurent Najman. Gradient Vector Fields of Discrete Morse Functions are Minimum Spanning Forests. 2022. ⟨hal-03614850v1⟩