Optimal point-set embedding of wheel graphs and a sub-class of 3-trees

Abstract

In the point-set embeddability problem, given a planar graph and a point set in the plane, one asks if there exists a straight-line drawing of the graph in the plane such that the nodes of the graph are mapped one-to-one onto the points of the set. By optimal embedding of a graph, we mean an embedding of minimum length or area. The point-set embeddability problem is NP-complete, and so far it has been solved polynomially only for a few classes of planar graphs. In this paper, we present optimal \(O(n\log {n})\)-time algorithms for embedding and \(O(n^2)\)-time algorithms for optimal embedding of wheel graphs and a sub-class of 3-trees.