The class of cubic Hamiltonian graphs is well studied class. I came across the fact that independent set problem is $NP$-complete when restricting input to cubic Hamiltonian graphs. I am interested in other hard problems on this class.
Which $NP$-complete problems on cubic graphs remain hard when restricted to cubic Hamiltonian graphs?
A survey of such hard problems would be very nice.
Given two NP-complete problems on a restricted graph class, I would like to understand the intractability boarderline when we further restrict input instances.
UPDATE: I am not interested in decision version of optimization problem such as clique problem (or maximum independent set). However, Dominating Clique problem is the kind of $NP$-complete problem that would interest me if it was hard on cubic Hamiltonian graphs. Dominating clique in graph $G(V,E)$ is a dominating set of $V$ and a clique of $G$.
Dominating clique problem is interesting for me because the problem definition does not contain a parameter ( unlike clique or MIS for which we must specify the size of the required solution).