Geodesic
Clustered Spanning Tree - Conditions for Feasibility
Discrete mathematics & theoretical computer science, ICGT 2018, Tome 21 (2019) no. 1.

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Let H =< V, S > be a hypergraph, where V is a set of vertices and S is a set of not necessarily disjoint clusters Si ⊆ V. The Clustered Spanning Tree problem is to find a spanning tree of G 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. 
@article{DMTCS_2019_21_1_a14,
     author = {Guttmann-Beck, Nili and Sorek, Zeev and Stern, Michal},
     title = {Clustered {Spanning} {Tree} - {Conditions} for {Feasibility}},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
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     number = {1},
     year = {2019},
     doi = {10.23638/DMTCS-21-1-15},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.23638/DMTCS-21-1-15/}
}
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Guttmann-Beck, Nili; Sorek, Zeev; Stern, Michal. Clustered Spanning Tree - Conditions for Feasibility. Discrete mathematics & theoretical computer science, ICGT 2018, Tome 21 (2019) no. 1. doi : 10.23638/DMTCS-21-1-15. http://geodesic.mathdoc.fr/articles/10.23638/DMTCS-21-1-15/

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