  We introduce a framework for mapping NP-Hard problems to adiabatic quantum
computing (AQC) architectures that are heavily restricted in both connectivity
and dynamic range of couplings, for which minor-embedding -- the standard
problem mapping method -- cannot be directly applied. Separating the mapping
into two distinct stages, we introduce problem-specific reductions for both
quadratic unconstrained binary optimisation (QUBO) and satisfiability (SAT) and
develop the subdivision-embedding method that is suitable for directly
embedding onto these heavily restricted architectures. The theory underpinning
this framework provides tools to aid in the manipulation of Ising Hamiltonians
for the purposes of Ising energy minimisation, and could be used to assist in
developing and optimising further problem mapping techniques. For each of the
problem mapping methods presented, we examine how the physical qubit count
scales with problem size on architectures of varying connectivity.
