  For each $c\ge 1$ we prove tight lower bounds on face sizes that must be
present to allow $1$- or $2$-cuts in simple duals of $c$-connected maps. Using
these bounds, we determine the smallest genus on which a $c$-connected map can
have a simple dual with a $2$-cut and give lower and some upper bounds for the
smallest genus on which a $c$-connected map can have a simple dual with a
$1$-cut.
