  We perform ray-tracing simulations evaluating the effect of a cD galaxy on
the strong lensing properties of five galaxy cluster halos obtained from N-body
simulations. The cD galaxy is modelled using both axially symmetric and
elliptical models and assuming several masses for its dark matter halo. The
effect of the cD orientation with respect to the mass distribution of the host
galaxy cluster is also investigated. The simulations are carried out in an open
and a flat model universe with cosmological constant. We find that the
enhancement of the cluster lensing cross sections for long and thin arcs due to
the presence of a massive cD at the cluster centre is typically less than 100%,
depending on the model used for the cD galaxy and its orientation. The impact
of the cD on the cluster efficiency for producing radially magnified images is
larger only for those clusters whose lensing cross section for radial arcs is
very small in absence of the central galaxy. We conclude that the presence of a
cD galaxy at the cluster centre can only moderately influence the cluster
efficiency for strong lensing and in particular fails to explain the
discrepancy between the observed number of giant arcs in galaxy clusters and
their abundance predicted from lensing simulations in the currently most
favoured $\Lambda$CDM model.
