An analogue of Morse theory for planar linear networks and the generalized Steiner problem

A study is made of the generalized Steiner problem: the problem of finding all the locally minimal networks spanning a given boundary set (terminal set). It is proposed to solve this problem by using an analogue of Morse theory developed here for planar linear networks. The space $\mathscr K$ of all planar linear networks spanning a given boundary set is constructed. The concept of a critical point and its index is defined for the length function $\ell$ of a planar linear network. It is shown that locally minimal networks are local minima of $\ell$ on $\mathscr K$ and are critical points of index 1. The theorem is proved that the sum of the indices of all the critical points is equal to $\chi(\mathscr K)=1$. This theorem is used to find estimates for the number of locally minimal networks spanning a given boundary set.