Geodesic
One-dimensional minimal fillings with negative edge weights
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 5 (2012), pp. 3-8

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A. O. Ivanov and A. A. Tuzhilin started an investigation of a particular case of Gromov Minimal Fillings problem generalization to the case of stratified manifolds using weighted graphs with nonnegative weight function as minimal fillings of finite metric spaces. In this paper we introduce generalized minimal fillings, i.e., minimal fillings where the weight function is not necessarily nonnegative. We prove that for any finite metric space its minimal filling has the minimum weight in the class of its generalized fillings. 
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     author = {A. O. Ivanov and Z. N. Ovsyannikov and N. P. Strelkova and A. A. Tuzhilin},
     title = {One-dimensional minimal fillings with negative edge weights},
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A. O. Ivanov; Z. N. Ovsyannikov; N. P. Strelkova; A. A. Tuzhilin. One-dimensional minimal fillings with negative edge weights. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 5 (2012), pp. 3-8. http://geodesic.mathdoc.fr/item/VMUMM_2012_5_a0/