Recently I am learning about solution of the 2-satiability problem using strongly connected components. There is a theorem related to this problem given below:
Let $F = Q_lx_1 Q_2x_2\ldots Q_nx_n C$ be a quantified Boolean formula with no free variables, where each $Q_i$ is either universal or existential, and $C$ is in conjunctive normal form. That is, $C$ is a conjunction of clauses, each clause is a disjunction of literals, and each literal is either a variable, $x_i$, or the negation of a variable, $x_i$ ($1 \leq i \leq n$).
Theorem 2: The formula $F$ is true if and only if none of the following three conditions holds:
3(i) An existential vertex $u$ is in the same strong component as its complement $u’$.
3(ii) A universal vertex $u_i$ is in the same strong component as an existential vertex $u_j$ such that $j < i$ (i.e., $x_i$ is not quantified within the scope of $Q_i$).
3(iii) There is a path from a universal vertex $u$ to another universal vertex $v$. (This condition includes the case that $v = u$.)
In the proof they use this definition: We call a vertex universal if the corresponding variable is universally quantified and existential otherwise.
But I am not familiar with the terms universal vertex and existential vertex. Can any one please explain these terms to me?