Outerplanarity of line graphs and iterated line graphs

The outerplanar index of a graph G is the smallest integer k such that the k th iterated line graph of G 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.