  In this work we analytically explain the origin of the mobility edge in the
partially disordered random regular graphs of degree d, i.e., with a fraction
$\beta$ of the sites being disordered, while the rest remain clean. It is shown
that the mobility edge in the spectrum survives in {a certain range of
parameters} $(d,\beta)$ at infinitely large uniformly distributed disorder. The
critical curve separating extended and localized states is derived analytically
and confirmed numerically. The duality in the localization properties between
the sparse and extremely dense RRG has been found and understood.
