On complexity of subset interconnection designs

Given a setX and subsetsX1,...,Xm, we consider the problem of finding a graphG with vertex setX and the minimum number of edges such that fori=1,...,m, the subgraphGi; induced byXi is connected. Suppose that for anyα pointsx1,...,xαε X, there are at mostβXi 's containing the set {x1,...,xα}. In the paper, we show that the problem is polynomial-time solvable for (α ⩽ 2,β ⩽ 2) and is NP-hard for (α⩾3,β=1), (α=l,β⩾6), and (α⩾2,β⩾3).