Outerplanarity of line graphs and iterated line graphs

The outerplanar index of a graph GG is the smallest integer kk such that the kkth iterated line graph of GG is non-outerplanar. In this note, we show: (i) the characterization of the forbidden subgraphs for graphs with outerplanar line graphs; (ii) that the outerplanar index of a graph is either infinite or at most 3. Furthermore, we give a full characterization of all graphs with respect to their outerplanar index.