A short proof of Oblakov's theorem

Abstract

We give a short proof that for a given planar set of terminals $P$ there is at most one locally minimal tree with prescribed directions at $P$ (i.e. two locally minimal trees cannot coincide in $B_\varepsilon(P)$). The new ingredient is a combinatorial result which says that a certain embedding of a bipartite cubic graph has the same numbers of balanced and unbalanced vertices.