  Dirac and Lov\'{a}sz independently characterized the $3$-connected graphs
with no pair of vertex-disjoint cycles. Equivalently, they characterized all
$3$-connected graphs with no prism-minors. In this paper, we completely
characterize the $3$-connected graphs with an edge that is contained in the
union of no pair of vertex-disjoint cycles. As applications, we answer the
analogous questions for edge-disjoint cycles and for $4$-connected graphs and
we completely characterize the $3$-connected graphs with no prism-minor using a
specified edge.
