Embedding Plane 3-Trees in ℝ2 and ℝ3
A point-set embedding of a planar graph G with n vertices on a set P of n points in ℝ d , d ≥ 1, is a straight-line drawing of G, where the vertices of G are mapped to distinct points of P. The problem of computing a point-set embedding of G on P is NP-complete in ℝ2, even when G is 2-outerplanar and the points are in general position. On the other hand, if the points of P are in general position in ℝ3, then any bijective mapping of the vertices of G to the points of P determines a point-set embedding of G on P. In this paper, we give an O(n 4/3 + ε )-expected time algorithm to decide whether a plane 3-tree with n vertices admits a point-set embedding on a given set of n points in general position in ℝ2 and compute such an embedding if it exists, for any fixed ε>0. We extend our algorithm to embed a subclass of 4-trees on a point set in ℝ3 in the form of nested tetrahedra. We also prove that given a plane 3-tree G with n vertices, a set P of n points in ℝ3 that are not necessarily in general position and a mapping of the three outer vertices of G to three different points of P, it is NP-complete to decide if G admits a point-set embedding on P respecting the given mapping.
KeywordsPlane Graph General Position Outer Face Outerplanar Graph Triangular Face
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