 Connected vertex cover

A vertex cover $C$ whose induced subgraph $G[C]$ is connected. The optimization problem minimizes $|C|$ subject to both conditions. A graph with edges in two distinct connected components of a graph has no connected vertex cover: covering each of these edges requires selected vertices in different components. Isolated vertices can be ignored when exactly one component contains all the edges. The edge-free case depends on whether the empty solution is admitted.