On the criteria of D-planarity of a tree

Abstract

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 $\phi (T)\subseteq D^2$, $\phi (T)\cap \partial D^2=\phi (V^{\ast })$ and if $♯V^{\ast }\ge 3$ then a cyclic order $\phi (C)$ of $\phi (V^{\ast })$ 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