  Given a graph $G$, the Connected Vertex Cover problem (CVC) asks to find a
minimum cardinality vertex cover of $G$ that induces a connected subgraph. In
this paper we describe some approaches to solve the CVC problem exactly. First,
we give compact mixed-integer extended formulations for CVC: these are the
first formulations proposed for this problem, and can be easily adapted to
variations of the problem such as Tree Cover. Second, we describe a simple
branch and bound algorithm for the CVC problem. Finally, we implement our
algorithm and compare its performance against our best formulation: contrary to
what usually happens for the classical Vertex Cover problem, our formulation
outperforms the branch and bound algorithm.
