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The rooted tree embedding problem into points in the plane. (English) Zbl 0791.05027
Let $$T$$ be a rooted tree with $$n$$ vertices and $$N$$ a set $$\{p_ 1,\dots,p_ n\}$$ of $$n$$ points in general position in $$\mathbb{R}^ 2$$. Denote the sets of vertices and edges of $$T$$ by $$V(T)$$ and $$E(T)$$, respectively. A bijection $$\varphi$$ from $$V(T)$$ to $$N$$ satisfying the conditions:
(C1) $$\varphi(r_ 1)=p_ 1$$ $$(r_ 1$$ is the root of $$T)$$,
(C2) for nonadjacent edges $$u_ 1v_ 1$$, $$u_ 2v_ 2 \in E(T)$$, the line segments $$\overline {\varphi (u_ 1) \varphi(v_ 1)}$$, $$\overline {\varphi (u_ 2) \varphi (v_ 2)}$$ are disjoint,
is called a rooted tree embedding (or $$rt$$-embedding).
The main theorem states that an $$rt$$-embedding of $$V(T)$$ on $$N$$ always exists and that some $$rt$$-embedding can be constructed in polynomial time with respect to $$n$$.

##### MSC:
 05C10 Planar graphs; geometric and topological aspects of graph theory 05C05 Trees 68U05 Computer graphics; computational geometry (digital and algorithmic aspects)
##### Keywords:
algorithm; rooted tree; embedding; polynomial time
Full Text:
##### References:
  Edelsbrunner, H. (1987),Algorithms in Combinatorial Geometry, Springer-Verlag, Berlin.  Pach, J.; Töröcsik, J.; Trotter, W. T. (ed.), Layout of rooted trees, 131-137, (1993), Providence, RI · Zbl 0792.05083  Perles, M. (1990), Open problem proposed at the DIMACS Workshop on Arrangements, Rutgers University.  Preparata, F. P., and Shamos, L. I. (1985),Computational Geometry—An Introduction, Springer-Verlag, New York. · Zbl 0759.68037
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