Abstract
This paper presents a new and efficient algorithm, IligraLIGRA, for inverse line graph construction. Given a line graph H, ILIGRA constructs its root graph G with the time complexity being linear in the number of nodes in H. If ILIGRA does not know whether the given graph H is a line graph, it firstly assumes that H is a line graph and starts its root graph construction. During the root graph construction, ILIGRA checks whether the given graph H is a line graph and ILIGRA stops once it finds H is not a line graph. The time complexity of ILIGRA with line graph checking is linear in the number of links in the given graph H. For sparse line graphs of any size and for dense line graphs of small size, numerical results of the running time show that ILIGRA outperforms all currently available algorithms.
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Ahn, Y.Y., Bagrow, J.P., Lehmann, S.: Link communities reveal multiscale complexity in networks. Nature 466(7307), 761–764 (2010)
Barabási, A.L., Albert, R.: Emergence of scaling in random networks. Science 286(5439), 509–512 (1999)
Bollobás, B.: Random Graphs. Cambridge University Press, Cambridge (2001)
Cauchy, A.L.: Cours d’analyse de l’Ecole Royale Polytechnique, vol. 3 (1821). Imprimerie royale, Paris (reissued by Cambridge University Press), Cambridge (2009)
Cvetković, D., Rowlinson, P., Simić, S.: Spectral Generalizations of Line Graphs. Cambridge University Press, Cambridge (2004)
Degiorgi, D.G., Simon, K.: A dynamic algorithm for line graph recognition. In: Proceedings of 21st International Workshop on Graph-Theoretic Concepts in Computer Science (Lecture Notes in Computer Science 1017), pp. 37–48. Springer-Verlag (1995)
Erdős, P., Rényi, A.: On random graphs, I. Publ. Math. (Debr.) 6, 290–297 (1959)
Evans, T., Lambiotte, R.: Line graphs, link partitions, and overlapping communities. Phys. Rev. E 80(1), 016105 (2009)
Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities, 2nd edn. Cambridge University Press, Cambridge (1988)
Krausz, J.: Démonstration nouvelle d’un théorème de Whitney sur les réseaux. Mat. Fiz. Lapok 50, 75–85 (1943)
Krawczyk, M.J., Muchnik, L., Manka-Krason, A., Kulakowski, K.: Line graphs as social networks. Phys. A 390, 2611–2618 (2011)
Lehot, P.G.H.: An optimal algorithm to detect a line graph and output its root graph. J. ACM 21, 569–575 (1974)
Manka-Krason, A., Kulakowski, K.: Assortativity in random line graphs. Acta Phys. Pol. B Proc. Suppl. 3(2), 259–266 (2010)
Manka-Krason, A., Mwijage, A., Kulakowski, K.: Clustering in random line graphs. Comput. Phys. Commun. 181(1), 118–121 (2010)
Mehlhorn, K., Näher, S.: LEDA: A Platform for Combinatorial and Geometric Computing. Cambridge University Press, Cambridge (1999)
Nacher, J.C., Ueda, U., Yamada, T., Kanehisa, M., Akutsu, T.: Line graphs as social networks. BMC Bioinfo. 24(207), 2611–2618 (2004)
Nacher, J.C., Yamada, T., Goto, S., Kanehisa, M., Akutsu, T.: Two complementary representations of a scale-free network. Phys. A 349, 349–363 (2005)
Naor, J., Novick, M.B.: An efficient reconstruction of a graph from its line graph in parallel. J. Algoritm. 11, 132–143 (1990)
Ore, O.: Theory of Graphs, vol. 21. American Mathematical Society Colloquium Publications (1962)
Roussopoulos, N.D.: A max{m, n} algorithm for detecting the graph h from its line graph g. Info. Process. Lett. 2, 108–112 (1973)
Simić, S.: An algorithm to recognize a generalized line graphs and ouput its root graph. Publ. Math. Inst. (Belgrade) 49(63), 21–26 (1990)
Van Mieghem, P.: Graph Spectra for Complex Networks. Cambridge University Press, Cambridge (2011)
van Rooij, A.C.M., Wilf, H.S.: The interchange graph of a finite graph. Acta Math. Acad. Sci. Hung. 16, 263–269 (1965)
Whitney, H.: Congruent graphs and the connectivity of graphs. Am. J. Math. 54, 150–168 (1932)
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This research was supported by Next Generation Infrastructures (Bsik).
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Liu, D., Trajanovski, S. & Van Mieghem, P. ILIGRA: An Efficient Inverse Line Graph Algorithm. J Math Model Algor 14, 13–33 (2015). https://doi.org/10.1007/s10852-014-9251-2
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DOI: https://doi.org/10.1007/s10852-014-9251-2