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The geometry of minimal networks with a given topology and a fixed boundary

In this paper we study the structure of the set of all locally minimal plane networks with a fixed topology and a fixed boundary . It is shown that if this set is non-empty, then it is a -dimensional convex body in the configuration space of the movable vertices of the network, where is the cyclomatic number for the movable subgraph in . In particular, all the networks in are parallel, have the same length, and can be deformed into one another in the class of locally minimal networks of the same type and with the same boundary. Moreover, we describe how two networks belonging to can be distinguished.