  Connected Vertex Cover is one of the classical problems of computer science,
already mentioned in the monograph of Garey and Johnson. Although the
optimization and decision variants of finding connected vertex covers of
minimum size or weight are well studied, surprisingly there is no work on the
enumeration or maximum number of minimal connected vertex covers of a graph. In
this paper we show that the maximum number of minimal connected vertex covers
of a graph is at most 1.8668^n, and these can be enumerated in time
O(1.8668^n). For graphs of chordality at most 5, we are able to give a better
upper bound, and for chordal graphs and distance-hereditary graphs we are able
to give tight bounds on the maximum number of minimal connected vertex covers.
