Are there "easier" or "harder" (sets of) instances of an $NP$-complete problem?
If yes (which i assume), how a polynomial-time mapping between $NP$-complete problems maps between easier or harder (sets of) instances?
A conjecture (related to directions of previous questions) is that there should be considerable mapping betwen "easier" to "harder" and "harder" to "easier" (if i may, with probability almost 1).
PS. This in effect tries to study (a little more in depth) the role of polynomial-time mappings used in $NP$-complete reductions.