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 $\varphi : T \to \rr^{2}$ which satisfies the following conditions $\varphi(T) \subseteq D^{2}$, $\varphi(T) \cap \partial D^{2} = \varphi(V^{\ast})$ and if $\sharp V^{\ast} \geq 3$ then a cyclic order $\varphi(C)$ of $\varphi(V^{\ast})$ coincides with a cyclic order which is generated by the orientation of $\partial D^{2} \cong S^{1}$.