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ILIGRA: an efficient inverse line graph algorithm. (English) Zbl 1347.05239
Summary: This paper presents a new and efficient algorithm, Iligra, 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.

##### MSC:
 05C85 Graph algorithms (graph-theoretic aspects) 05C76 Graph operations (line graphs, products, etc.) 68Q25 Analysis of algorithms and problem complexity
##### Keywords:
graph algorithm; line graph; root graph
ILIGRA; LEDA
Full Text:
##### References:
 [1] Ahn, YY; Bagrow, JP; Lehmann, S., Link communities reveal multiscale complexity in networks, Nature, 466, 761-764, (2010) [2] Barabási, AL; Albert, R., Emergence of scaling in random networks, Science, 286, 509-512, (1999) · Zbl 1226.05223 [3] Bollobás, B.: Random Graphs. Cambridge University Press, Cambridge (2001) · Zbl 0979.05003 [4] Cauchy, A.L.: Cours d’analyse de l’Ecole Royale Polytechnique, vol. 3 (1821). Imprimerie royale, Paris (reissued by Cambridge University Press), Cambridge (2009) · Zbl 1205.26006 [5] Cvetković, D., Rowlinson, P., Simić, S.: Spectral Generalizations of Line Graphs. Cambridge University Press, Cambridge (2004) · Zbl 1061.05057 [6] 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) [7] Erdős, P.; Rényi, A., On random graphs, I, Publ. Math. (Debr.), 6, 290-297, (1959) · Zbl 0092.15705 [8] Evans, T., Lambiotte, R.: Line graphs, link partitions, and overlapping communities. Phys. Rev. E 80(1), 016105 (2009) [9] Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities, 2nd edn. Cambridge University Press, Cambridge (1988) [10] Krausz, J., Démonstration nouvelle d’un théorème de Whitney sur les réseaux, Mat. Fiz. Lapok, 50, 75-85, (1943) · Zbl 0061.41401 [11] Krawczyk, MJ; Muchnik, L.; Manka-Krason, A.; Kulakowski, K., Line graphs as social networks, Phys. A, 390, 2611-2618, (2011) [12] Lehot, PGH, An optimal algorithm to detect a line graph and output its root graph, J. ACM, 21, 569-575, (1974) · Zbl 0295.05118 [13] Manka-Krason, A.; Kulakowski, K., Assortativity in random line graphs, Acta Phys. Pol. B Proc. Suppl., 3, 259-266, (2010) [14] Manka-Krason, A.; Mwijage, A.; Kulakowski, K., Clustering in random line graphs, Comput. Phys. Commun., 181, 118-121, (2010) · Zbl 1202.05129 [15] Mehlhorn, K., Näher, S.: LEDA: A Platform for Combinatorial and Geometric Computing. Cambridge University Press, Cambridge (1999) · Zbl 0976.68156 [16] Nacher, JC; Ueda, U.; Yamada, T.; Kanehisa, M.; Akutsu, T., Line graphs as social networks, BMC Bioinfo., 24, 2611-2618, (2004) [17] Nacher, JC; Yamada, T.; Goto, S.; Kanehisa, M.; Akutsu, T., Two complementary representations of a scale-free network, Phys. A, 349, 349-363, (2005) [18] Naor, J.; Novick, MB, An efficient reconstruction of a graph from its line graph in parallel, J. Algoritm., 11, 132-143, (1990) · Zbl 0715.68070 [19] Ore, O.: Theory of Graphs, vol. 21. American Mathematical Society Colloquium Publications (1962) · Zbl 0105.35401 [20] Roussopoulos, ND, A maxm,n algorithm for detecting the graph h from its line graph g, Info. Process. Lett., 2, 108-112, (1973) · Zbl 0274.05116 [21] Simić, S., An algorithm to recognize a generalized line graphs and ouput its root graph, Publ. Math. Inst. (Belgrade), 49, 21-26, (1990) [22] Van Mieghem, P.: Graph Spectra for Complex Networks. Cambridge University Press, Cambridge (2011) · Zbl 1232.05128 [23] Rooij, ACM; Wilf, HS, The interchange graph of a finite graph, Acta Math. Acad. Sci. Hung., 16, 263-269, (1965) · Zbl 0139.17203 [24] Whitney, H., Congruent graphs and the connectivity of graphs, Am. J. Math., 54, 150-168, (1932) · Zbl 0003.32804
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