Line Graphs and Forbidden Induced Subgraphs

Beineke and Robertson independently characterized line graphs in terms of nine forbidden induced subgraphs. In 1994, Šoltés gave another characterization, which reduces the number of forbidden induced subgraphs to seven, with only five exceptional cases. A graph is said to be a dumbbell if it consists of two complete graphs sharing exactly one common edge. In this paper, we show that a graph with minimum degree at least seven that is not a dumbbell is a line graph if and only if it does not contain three forbidden induced subgraphs including K1, 3 and K5−e. Applications of our main results to other forbidden induced subgraph characterizations of line graphs and to hamiltonian line graphs are also discussed.