A straight-line drawing of a plane graph G is a planar drawing of G, where each vertex is drawn as a point and each edge is drawn as a straight-line segment. Given a set S of n points on the Euclidean plane, a point-set embedding of a plane graph G with n vertices on S is a straight-line drawing of G, where each vertex of G is mapped to a distinct point of S. The problem of deciding if G admits a point-set embedding on S is NP-complete in general and even when G is 2-connected and 2-outerplanar. In this paper we give an O(n2log n) time algorithm to decide whether a plane 3-tree admits a point-set embedding on a given set of points or not, and find an embedding if it exists. We prove an Ω(n log n) lower bound on the time complexity for finding a point-set embedding of a plane 3-tree. Moreover, we consider a variant of the problem where we are given a plane 3-tree G with n vertices and a set S of k > n points, and give a polynomial time algorithm to find a point-set embedding of G on S if it exists. © 2011 Springer-Verlag.
CITATION STYLE
Nishat, R. I., Mondal, D., & Rahman, M. S. (2011). Point-set embeddings of plane 3-trees (extended abstract). In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 6502 LNCS, pp. 317–328). https://doi.org/10.1007/978-3-642-18469-7_29
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