the hamiltonian circuit (HC) problem remains NP-complete when restricted to chordal bipartite graphs (cbg) and becomes polynomial when restricted further to bipartite distance hereditary graphs (bdhg). I'm not aware of any "well established" class of graphs between bdhg and cbg. If there is one, you could consider the computational complexity of HC when restricted to this class. Do you have any such class of graphs in mind?
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