On triangulating kk-outerplanar graphs

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