  We show that every quadrangulation of the sphere can be transformed into a
$4$-cycle by deletions of degree-$2$ vertices and by $t$-contractions at
degree-$3$ vertices. A $t$-contraction simultaneously contracts all incident
edges at a vertex with stable neighbourhood. The operation is mainly used in
the field of $t$-perfect graphs. We further show that a non-bipartite
quadrangulation of the projective plane can be transformed into an odd wheel by
$t$-contractions and deletions of degree-$2$ vertices. We deduce that a
quadrangulation of the projective plane is (strongly) $t$-perfect if and only
if the graph is bipartite.
