  In this paper, we introduce relative LS category of a map and study some of
its properties. Then we introduce `higher topological complexity' of a map, a
homotopy invariant. We give a cohomological lower bound and compare it with
previously known `topological complexity' of a map. Moreover, we study the
relation between Lusternik-Schnirelmann category and topological complexity of
two closed oriented manifolds connected by a degree one map.
