##### Algorithm to solve 2-SAT

First decompose the implication graph into SCCs, and then if $x_i$ and $\neg x_i$ is in the same SCC, then the answer is impossible. Else it suffices to choose the variable that appears later in the topological order to be true.

##### Problems and Techniques

**Enforcing one variable to be true**: link

To do that, we can just add an edge from $\neg x_i$ to $x_i$

**Enforcing exactly one variable among several to be true**

This can be done in $O(n)$. Order the nodes arbitrarily. Then we can build prefix nodes for each prefix part of the nodes and establish implication between those prefix nodes and the variables themselves.