On the criteria of D-planarity of a tree
Abstract
Description
Let $T$ be a tree with a fixed subset of vertices $V^{\ast}$ such that there is a cyclic order $C$ on it and all terminal vertices are contained in this set. Let $D^{2} = \{(x,y) \in \rr^{2} | x^{2} + y^{2} \leq 1 \}$ be a closed oriented 2--dimensional disk. The tree $T$ is called D-planar if there exists an embedding $φ: T \to \rr^{2}$ which satisfies the following conditions $φ(T) \subseteq D^{2}$, $φ(T) \cap \partial D^{2} = φ(V^{\ast})$ and if $\sharp V^{\ast} \geq 3$ then a cyclic order $φ(C)$ of $φ(V^{\ast})$ coincides with a cyclic order which is generated by the orientation of $\partial D^{2} \cong S^{1}$.
LaTeX, 31 pages, 1 figure, definition of complete cyclic order was corrected
LaTeX, 31 pages, 1 figure, definition of complete cyclic order was corrected