Motion primitive-based (lattice-based) graphs have been used extensively in navigation, but application to high-dimensional state-spaces has remained limited by computational complexity. In this work, we show how these graphs can be applied to mobile manipulation. The formation of closed chains in tasks that involve contacts with the environment may reduce the number of available degrees of freedom but add complexity in terms of constraints in the high-dimensional state space. We propose a novel method to reduce dimensionality by abstracting away the constraints associated with closed-chain systems. Proofs are introduced for the application to graph-search and its theoretical guarantees of optimality. The dimensionality-reduction is done in a manner that enables finding optimal solutions to low-dimensional problems which map to correspondingly optimal full-dimensional solutions. We demonstrate the usefulness of our method with simulation results; we apply our approach to moving an object in 2D using a mobile manipulation platform with a planar arm.