Plane 3-trees: Embeddability & Approximation

We give anO(n log n)-time linear-space algorithm that, given a plane 3-tree G with n vertices and a set S of n points in the plane, determines whether G has a point-set embedding on S (i.e., a planar straight-line drawing of G where each vertex is mapped to a distinct point of S), improving the O(n)-time O(n)-space algorithm of Moosa and Rahman. Given an arbitrary plane graph G and a point set S, Di Giacomo and Liotta gave an algorithm to compute 2-bend pointset embeddings of G on S using O(W ) area, where W is the length of the longest edge of the bounding box of S. Their algorithm uses O(W ) area even when the input graphs are restricted to plane 3-trees. We introduce new techniques for computing 2-bend point-set embeddings of plane 3-trees that takes only O(W ) area. We also give approximation algorithms for point-set embeddings of plane 3-trees. Our results on 2bend point-set embeddings and approximate point-set embeddings hold for partial plane 3-trees (e.g., series-parallel graphs and Halin graphs).