In Cormen's Algorithms book on NP-completeness they prove various problems are NP-complete by reducing a previously proved NP-complete problem (call $K$) to current problem (call $L$). Each proof involves some clever construction which reduces all instances of $K$ to few instances of $L$. Here is the proof order they follow. CIRCUIT-SAT, SAT, 3 CNF-SAT, CLIQUE, VERTEX-COVER, HAM-CYCLE, TSP. e.g. in reducing VERTEX-COVER to HAM-CYCLE they use a widget which does the trick.
After this previous question of mine, I think one can reduce back. i.e. one can reduce HAM-CYCLE to VERTEX-COVER problem. I tried searching web for such reductions, but most of the link return the normal reduction order. I'm interested to see if one can reduce in reverse order. i.e. TSP to HAM-CYCLE to VERTEX-COVER to CLIQUE to 3 CNF-SAT to SAT
I'm looking for reverse constructive proofs. I know all of these problems belong to NP-complete hence equivalent.
You don't have to give complete proof as an answer. Proof sketches are fine too. If you can point me where these proofs are available online, that's completely fine too. I'm just trying to lean how constructions are leveraged among problems that look so different on surface. Thanks!