Point-set embeddings of plane 3-trees

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 in 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(n2 ) 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. We then 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 present a dynamic programming algorithm to find a point-set embedding of ...

• Type: Journal
• Year: 2012
• Where: COMGEO
• Authors: Rahnuma Islam Nishat, Debajyoti Mondal, Md. Saidur Rahman
• Keywords: Applied Computing, COMGEO 2012, Dynamic Programming Algorithm, Plane Graph, Straight Line Segment