A study on H-line graphs

For a connected graph H of order at least 3, the H-line graph of a graph is defined as that graph whose vertices are the edges of G and where two vertices are adjacent if and only if the corresponding edges of G are adjacent and belong to a common copy of H . In particular, when H = P3, the H-line graph HL(G) is the standard line graph L(G) and for any connected graph G on at least three vertices, GL(G) = L(G). For k ≥ 2, the k iterated H-line graph HL(G) is defined as HL(HLk−1(G)), where HL(G) = HL(G) and HLk−1(G) is assumed to be non-empty. Chartrand et al. characterized those graphs for which the sequence {HLk(G)} converges, when H is P4, P5 or K1,n, n ≥ 3. In this paper we characterize those graphs G for which the sequence {HLk(G)} converges, when H is P6.