Clustered Spanning tree - Conditions for feasibility

Let H =< V, S > be a hypergraph, V is a set of vertices and S = {S₁, ⋯, S_m} is a set of not necessarily disjoint clusters S_i ⊆ V such that [∪^m _i=1S_i = V. The Clustered Spanning Tree problem is to find a tree spanning all the vertices of V which satisfies that each cluster induces a subtree, when it exists. We provide an efficient and unique algorithm which finds a feasible solution tree for H when it exists, or states that no feasible solution exists. The paper also uses special structures of the intersection graph of H to construct a feasible solution more efficiently. For cases when the hypergraph does not have a feasible solution tree, we consider adding vertices to exactly one cluster in order to gain feasibility. We characterize when such addition can gain feasibility, find the appropriate cluster and a possible set of vertices to be added.