We give a precise definition of a net as a geometric realization of a graph later on; now, in the introduction, by a net in a metric space X we mean any connected nonempty set N ? X representable as a finite union of rectifiable embedded curves with pairwise disjoint interiors. Here, an embedded curve is the image of the closed interval under a continuous mapping injective on the interior (an open interval). The length of a net is defined as the sum of lengths of the curves forming this net. Clearly, the length of any net is well defined, i.e., does not depend on the partition of the net into curves. Let D be a finite set of points in X. A net N_0 is said to be a shortest net with boundary D in X if D ? N_0 and the length any net N' in X for which D ? N' is at least the length of N_0. A net N_0 is called a locally minimal net with boundary D in X if D ? N_0 and, for any point P ∈ N_0 and any sufficiently small closed neighborhood B_P of this point such that the set N_0 ∩ B_P is connected, this set N_0 ∩ B_P is a shortest net with boundary (D ∩ B_P) ∪ (?BP ∩ N_0), where ?B_P denotes the boundary of the neighborhood B_P.
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