On embedding an outer-planar graph in a point set

Abstract

Given an n-vertex outer-planar graph G and a set P of n points in the plane, we present an O( nlog 3 n) time and O( n) space algorithm to compute a straight-line embedding of G in P, improving upon the algorithm in [8,12] that requires O( n 2) time. Our algorithm is near-optimal as there is an Ω(n logn) lower bound for the problem [4]. We present a simpler O( nd) time and O( n) space algorithm to compute a straight-line embedding of G in P where log n⩽ d⩽2 n is the length of the longest vertex disjoint path in the dual of G. Therefore, the time complexity of the simpler algorithm varies between O( nlog n) and O( n 2) depending on the value of d. More efficient algorithms are presented for certain restricted cases. If the dual of G is a path, then an optimal Θ( nlog n) time algorithm is presented. If the given point set is in convex position then we show that O( n) time suffices.