The MST-fan of a regular subdivision

TL;DR

This paper studies the structure of minimum spanning trees in the dual graph of a regular triangulation, revealing a polyhedral fan organization and connections to tropical geometry, with applications in data signal detection.

Contribution

It introduces the MST-fan, a new polyhedral organization of the parameter space for minimum spanning trees in regular subdivisions, linking it to tropical geometry and matroid theory.

Findings

The MST-fan subdivides the secondary cone into parameter cones.

Partial description of the local face structure of the MST-fan.

Connections established between the MST-fan and tropical geometry via matroids and Bergman fans.

Abstract

The dual graph $\Gamma(h)$ of a regular triangulation $\Sigma(h)$ carries a natural metric structure. The minimum spanning trees of $\Gamma(h)$ recently proved to be conclusive for detecting significant data signal in the context of population genetics. In this paper we prove that the parameter space of such minimum spanning trees is organized as a polyhedral fan, called the MST-fan of $\Sigma(h)$, which subdivides the secondary cone of $\Sigma(h)$ into parameter cones. We partially describe its local face structure and examine the connection to tropical geometry in virtue of matroids and Bergman fans.