Steiner connectivity problems in hypergraphs
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We say that a tree T is an S-Steiner tree if S ⊆ V(T) and a hypergraph is an S-Steiner hypertree if it can be trimmed to an S-Steiner tree. We prove that it is NP-hard to decide, given a hypergraph ℋ and some S ⊆ V(ℋ), whether there is a subhypergraph of ℋ which is an S-Steiner hypertree. As corollaries, we give two negative results for two Steiner orientation problems in hypergraphs. Firstly, we show that it is NP-hard to decide, given a hypergraph ℋ, some r ∈ V(ℋ) and some S ⊆ V(ℋ), whether this hypergraph has an orientation in which every vertex of S is reachable from r. Secondly, we show that it is NP-hard to decide, given a hypergraph ℋ and some S ⊆ V(ℋ), whether this hypergraph has an orientation in which any two vertices in S are mutually reachable from each other. This answers a longstanding open question of the Egerváry Research group. On the positive side, we show that the problem of finding a Steiner hypertree and the first orientation problem can be solved in polynomial time if the number of terminals |S| is fixed.
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