Finding induced trees

Let G=(V(G),E(G))G=(V(G),E(G)) be a finite connected undirected graph and W⊂V(G)W⊂V(G) a subset of vertices. We are searching for a subset X⊂V(G)X⊂V(G) such that W⊂XW⊂X and the subgraph induced on XX is a tree. NPNP-completeness results and polynomial time algorithms are given assuming that the cardinality of WW is fixed or not. Besides we give complexity results when XX has to induce a path or when GG is a weighted graph. We also consider problems where the cardinality of XX has to be minimized.