P vs NP

We’ve reached the NP-completeness section of my Fundamental Algorithms class, and I’ve noticed something interesting about P and NP.

class P NP-complete
Euler tour Hamiltonian Cycle
Shortest path Longest simple path

We have problems from a variety of different fields, for which one instance of the problem is in class P and the other is NP-Complete. I conjecture that whatever property it is that separates one class from the other, it must show up in all such fields. So the underlying fundamental reason for separating P from NP must translate from graph theory to boolean logic theory. It would be nice to gather such example divisions and then try to formulate an isomorphisms between fields. I’d think that eventually an isomorphism will be found that sheds enough light on the P vs NP problem that it can finally be cracked.

What really got me interested in this problem was noticing that Hamiltonian Cycle would be very easy to solve if we could convert it into an Euler tour by constructing an interchange graph; where all the nodes become edges and all the edges become nodes. But interchange is not a well defined operation on graphs, indeed it isn’t even possible in some cases. Example: The graph of 5 nodes and 0 edges. The interchange would have 0 nodes and 5 edges, which isn’t a graph. So, I was wildly speculating that perhaps what separates P from NP are these impossible operations. But to shed some light on the situation, I’d first want to construct a reduction of Euler tour from 2-SAT which mirrors the reduction of Hamiltonian cycle from 3-SAT.