  A k-outerplanar graph is a graph that can be drawn in the plane without
crossing such that after k-fold removal of the vertices on the outer-face there
are no vertices left. In this paper, we study how to triangulate a
k-outerplanar graph while keeping its outerplanarity small. Specifically, we
show that not all k-outerplanar graphs can be triangulated so that the result
is k-outerplanar, but they can be triangulated so that the result is
(k+1)-outerplanar.
