Recall that a vertex cover is a set of vertices where every edge of the graph is incident to at least one vertex in the set. How can you leverage this property to construct a dominating set?

Consider transforming the vertex cover instance into a dominating set instance by adding auxiliary vertices and edges. How might you ensure that any dominating set in the new graph corresponds to a valid vertex cover in the original graph?

Explore the idea of creating a bipartite graph where one partition represents the original vertices and the other partition represents edges. How can you modify this structure to enforce the dominating set constraints while preserving the vertex cover properties?