Graphs whose line graphs are ring graphs

Given a graph H, a path of length at least two is called an H-path if meets H exactly in its ends. A graph G is a ring graph if each block of G which is not a bridge or a vertex can be constructed inductively by starting from a single cycle and then in each step adding an H-path that meets graph H in the previous step in two adjacent vertices. In this article, we classify all graphs whose line graphs and total graphs are ring graphs.