# Reduce Vertex cover to SAT

I need to reduce the vertex cover problem to a SAT problem, or rather tell whether a vertex cover of size k exists for a given graph, after solving with a SAT solver. I know how to reduce a 3-SAT problem to vertex cover problem, by constructing the subgraphs for each variable (x, !x) and for each clause (a triable). But I am not getting,how to do other way round?

I was thinking of first forming a DNF ,with electing k vertices at first and then convert it to a CNF, by enumerating all clauses. Is there any other method?

• Converting from DNF to CNF can give an exponential blow-up in the size of the formula. Unless you can guarantee that this won't happen for the DNFs you need to deal with, your reduction won't run in polynomial time. – David Richerby Mar 9 '14 at 20:45
• Here you go Amrith citeseerx.ist.psu.edu/viewdoc/… I hope that answers your question – user15610 Mar 12 '14 at 17:03

Let's introduce a variable $x_i$ for every node $i$, representing the condition that the node is part of the vertex cover. Then for every edge $\{v,w\}$ we introduce the clause $x_v \lor x_w$.
We have the additional condition $x_1 + x_2 + \ldots \ +\ x_n \leq k$. We can combine an addition circuit to compute the bits of the sum with a comparator circuit to enforce the inequality. The CNF of the resulting circuit has only polynomially many clauses.