Outerplanarity of line graphs and iterated line graphs

Abstract

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.