One-dimensional minimal fillings with negative edge weights
  Ivanov and Tuzhilin started an investigation of a particular case of Gromov Minimal Fillings problem (generalized to the case of stratified manifolds). Weighted graphs with non-negative weight function were used as minimal fillings of finite metric spaces. In the present paper we introduce generalized minimal fillings, i.e. minimal fillings where the weight function is not necessarily non-negative. We prove that for any finite metric space its minimal filling has the minimum weight in the class of all generalized fillings of the space. 
Alexandr O. Ivanov, Zachar N. Ovsyannikov, Natalia P. Strelkova,   Alexey A. Tuzhilin
math.MG