Proof:

- 1.
- Given C, check that all elements of X are members of some set in C and
that
.
- 2.
- L' = VC
- 3.
- Given G, k
VC, define F such that each element of F is a
subset for a vertex v in G containing v and all vertices reachable by an edge
from v.

Let X = V. Then X,F, k SC. - 4.
- If C is the vertex cover of G, k
VC, then
every vertex u in G is incident from an edge (u,v) where either u
C
or v
C. Thus all vertices will appear in some set in F, and the sets
in F corresponding to the vertices in C make up the set covering of
X, F, k
SC.