How to reduce MAX-2SAT problem to finding a cut in a graph
Idea: Have two vertices $x,\bar{x}$ for each variable, and connect them with an edge with some large weight which will force any maximal cut to put them in different parts of the cut. The idea is that for any two variables $x,y$, $x$ and $y$ get the same truth value iff the vertices $x,y$ are in the same part of the cut. This way you encode a truth assignment.

For each clause $x \lor y$, we would like to do something of the following sort: add a vertex $\varphi_{x \lor y}$ and connect it to both $x$ and $y$. In order to guarantee that $\varphi_{x \lor y}$ is in some specific part of the cut, add a new vertex and connect it to all "clause" vertices with an edge having some large weight. This idea doesn't quite work since the value of the cut depends on how many literals satisfy the clause. Find a way to fix it.