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Given a class of (undirected) graphs C, we say that a FAS F is a C-FAS if the graph induced by the edges of F (forgetting their orientations) belongs to C. We show that deciding if a tournament has a C-FAS is NP-complete when C is the class of all forests. We are motivated by connections between C-FAS and structural parameters of tournaments, such as the dichromatic number, the clique number of tournaments, and the strong Erd{o}s-Hajnal property.
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Aboulker et al. (2024) studied this question.
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