Compactifications of phylogenetic systems and electrical networks

Abstract

We demonstrate a graphical map, a new correspondence between circular electrical networks and circular planar split systems. When restricted to the planar circular electrical case, this graphical map finds the split system associated uniquely to the Kalmanson resistance distance of the dual network, matching the induced split system familiar from phylogenetics. This is extended to compactifications of the respective spaces, taking cactus networks to the newly defined compactified split systems. The graphical map preserves both components and cactus structure, allowing an elegant enumeration of induced phylogenetic split systems.