Equivalence of Four Descriptions of Generalized Line Graphs

A new characterization of generalized line graphs, analogous to that of line graphs found by Van Rooij and Wilf [Acta Math Acad Sci Hungar 16 (1965), 263-269] is obtained. By a cycle of implications, we settle the equivalence of the definition of generalized line graph given by Hoffman [Combinatorial Structures and their Applications, Gordon and Breach, NY, 1970, pp. 173-176], its description by vectorial representation [P. J. Cameron et al., Algebra 43 (1976), 305-327], the current characterization and the one involving minimal nongeneralized line graphs [D. Cvetkovic et al., J Graph Theory 5(4) (1981), 385-399; D. Cvetkovic et al., Des Codes Cryptogr 34(2-3) (2005), 229-240; S. B. Rao et al., Combinatorics and Graph Theory, Lecture Notes 885, Springer, Berlin, 1981, pp. 459-472; G. R. Vijayakumar, J Combin Inform System Sci 9(3) (1984), 182-192]. This process includes a short method of finding the aforementioned graphs, analogous to the method found by Beineke [J Combin Theory 9 (1970), 129-135] to obtain all minimal nonline graphs. (C) 2010 Wiley Periodicals, Inc. J Graph Theory 67: 27-33, 2011