NP-Completeness of [formula omitted]-orientations for plane graphs

Abstract

An s t - orientation or bipolar orientation of a 2-connected graph G is an orientation of its edges to generate a directed acyclic graph with a single source s and a single sink t . Given a plane graph G and two exterior vertices s and t , the problem of finding an optimum s t -orientation, i.e., an s t -orientation in which the length of a longest s t - path is minimized, was first proposed indirectly by Rosenstiehl and Tarjan in and then later directly by He and Kao in . In this paper, we prove that, given a 2-connected plane graph G , two exterior vertices s , t , and a positive integer K , the decision problem of whether G has an s t -orientation, where the maximum length of an s t -path is ≤ K , is NP-Complete. This solves a long standing open problem on the complexity of optimum s t -orientations for plane graphs. As a by-product, we prove that the NP-Completeness result holds for planar graphs as well.