Abstract
An st-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 vertices s and t on the exterior face of G, the problem of finding an optimum st-orientation, i.e., an st-orientation in which the length of the longest st-path is minimized, was first proposed indirectly by Rosenstiehl and Tarjan in [14] and then later directly by He and Kao in [6]. In this paper, we prove that, given a 2-connected plane graph G, two vertices s, t, on the exterior face of G and a positive integer K, the decision problem of whether G has an st-orientation, where the maximum length of an st-path is ≤ K, is NP-Complete. This solves a long standing open problem on the complexity of optimum st-orientations for plane graphs.KeywordsPlane GraphDirected PathHorizontal PathCorner EdgeCorner VertexThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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