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.
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Sadasivam, S., Zhang, H. (2009). NP-Completeness of st-Orientations for Plane Graphs. In: Kutyłowski, M., Charatonik, W., Gębala, M. (eds) Fundamentals of Computation Theory. FCT 2009. Lecture Notes in Computer Science, vol 5699. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03409-1_27
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