Point-Set Embeddings of Plane 3-Trees
- Cite this paper as:
- Nishat R.I., Mondal D., Rahman M.S. (2011) Point-Set Embeddings of Plane 3-Trees. In: Brandes U., Cornelsen S. (eds) Graph Drawing. GD 2010. Lecture Notes in Computer Science, vol 6502. Springer, Berlin, Heidelberg
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(n2logn) 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 logn) 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.