Forbidden Tournaments and the Orientation Completion Problem
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For a fixed finite set of finite tournaments ℱ, the ℱ-free orientation problem asks whether a given finite undirected graph G has an ℱ-free orientation, i.e., whether the edges of G can be oriented so that the resulting digraph does not embed any of the tournaments from ℱ. We prove that for every ℱ, this problem is in P or NP-complete. Our proof reduces the classification task to a complete complexity classification of the orientation completion problem for ℱ, which is the variant of the problem above where the input is a directed graph instead of an undirected graph, introduced by Bang-Jensen, Huang, and Zhu (2017). Our proof uses results from the theory of constraint satisfaction, and a result of Agarwal and Kompatscher (2018) about infinite permutation groups and transformation monoids.
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