Complete Problems for Monotone NP

We show that the problem of deciding whether a digraph has a Hamiltonian path between two specified vertices and the problem of deciding whether a given graph has a cubic subgraph are complete for monotone NP via monotone projection translations. We also show that the problem of deciding whether a uniquely partially orderable (resp. comparability) graph has a cubic subgraph is complete for NP via projection translations: these problems were previously not even known to be complete for NP via polynomial-time reductions (the class of uniquely partially orderable graphs is a proper subclass of the class of comparability graphs).