Fomin and Villanger (, STACS 2010) proved that MAXIMUM INDEPENDENT SET, FEEDBACK VERTEX SET, and more generally the problem of finding a maximum induced subgraph of treewith at most a constant t, can be solved in polynomial time on graph classes with polynomially many minimal separators. We extend these results in two directions. Let Gpoly be the class of graphs with at most poly(n) minimal separators, for some polynomial poly. We show that the odd powers of a graph G have at most as many minimal separators as G. Consequently, DISTANCE-d INDEPENDENT SET, which consists in finding maximum set of vertices at pairwise distance at least d, is polynomial on Gpoly, for any even d. The problem is NP-hard on chordal graphs for any odd d ≥ 3 . We also provide polynomial algorithms for CONNECTED VERTEX COVER AND CONNECTED FEEDBACK VERTEX SET on subclasses of Gpoly including chordal and circular-arc graphs, and we discuss variants of independent domination problems.