An analogue of Morse theory for planar linear networks and the generalized Steiner problem
Григорий Анатольевич Карпунин · Sbornik Mathematics · 2000
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 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 of a planar linear network. It is shown that locally minimal networks are local minima of on 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 . This theorem is used to find estimates for the number of locally minimal networks spanning a given boundary set.