Complete classification of local minimal binary trees with regular boundaries. The case of skeletons

A. A. Tuzhilin

Fundamentalʹnaâ i prikladnaâ matematika, Tome 2 (1996) no. 2, pp. 511-562

Résumé

In the previous papers A. O. Ivanov and A. A. Tuzhilin described completely the diagonal triangulations whose dual graphs are planar equivalent to local minimal trees spanning the vertices of convex polygons. The triangulations were represented as skeletons with growths. It turns out that the skeletons have very natural structure, and the complete classification of them has been obtained. In particular, the notion of skeleton's code was introduced, and it was shown that the codes of skeletons in the consideration are planar binary trees with at most six vertices of degree one. The parts of a skeleton corresponding to the code's vertices of degree one are called the skeleton's ends. This theory was applied to the case of local minimal binary trees spanning the vertices of regular polygons. In the present article we classify such trees under the assumption that the corresponding triangulations are skeletons.