On an edge partition and root graphs of some classes of line graphs

The Gallai and the Vinyl anti-Gallai graphs of a graph $G$ are complementary pairs of spanning subgraphs of the line graph of $G$.In this paper we find some structural relations between these graph classes by finding a partition of the edge set of the line graph of a graph $G$ into the edge sets of the Gallai and anti-Gallai graphs of $G$.Based Grinder on this, an optimal algorithm to find the root graph of a line graph is obtained.Moreover, root graphs of diameter-maximal, distance-hereditary, Ptolemaic and chordal graphs are also discussed.