Abstract
The degeneracy of an n-vertex graph G is the smallest number k such that every subgraph of G contains a vertex of degree at most k. We present an algorithm for enumerating all simple cycles of length p in an n-order k-degenerate graph running in time \(\mathcal {O}(n^{\lfloor {p/2} \rfloor } k^{\lceil p/2 \rceil })\). We then show that this algorithm is worst-case output size optimal by proving a \(\varTheta (n^{\lfloor {p/2} \rfloor } k^{\lceil {p}/{2} \rceil })\) bound on the maximal number of simple p-length cycles in these graphs. Our results also apply to induced (chordless) cycles.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Alon, N., Yuster, R., Zwick, U.: Finding and counting given length cycles. Algorithmica 17(3), 209–223 (1997)
Barabási, A.-L., Albert, R.: Emergence of scaling in random networks. Science 286(5439), 509–512 (1999)
Batagelj, V., Zaversnik, M.: An \(\cal{O}(m)\) algorithm for cores decomposition of networks (2003). arXiv preprint arXiv:cs/0310049
Birmelé, E., Ferreira, R., Grossi, R., Marino, A., Pisanti, N., Rizzi, R., Sacomoto, G.: Optimal listing of cycles and st-paths in undirected graphs, pp. 1884–1896 (2013)
Björklund, A., Kaski, P., Kowalik, Ł.: Counting thin subgraphs via packings faster than meet-in-the-middle time. In: Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 594–603. Society for Industrial and Applied Mathematics (2014)
Cai, L., Chan, S.M., Chan, S.O.: Random separation: a new method for solving fixed-cardinality optimization problems. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 239–250. Springer, Heidelberg (2006). doi:10.1007/11847250_22
Chiba, N., Nishizeki, T.: Arboricity and subgraph listing algorithms. SIAM J. Comput. 14(1), 210–223 (1985)
Chrobak, M., Eppstein, D.: Planar orientations with low out-degree and compaction of adjacency matrices. Theor. Comput. Sci. 86(2), 243–266 (1991)
Dorn, F.: Planar subgraph isomorphism revisited. In: 27th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Dagstuhl, Germany, vol. 5, pp. 263–274. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik (2010)
Eppstein, D.: Subgraph isomorphism in planar graphs and related problems. In: Proceedings of the Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 1995, Philadelphia, PA, USA, pp. 632–640. Society for Industrial and Applied Mathematics (1995)
Giscard, P.-L., Kriege, N., Wilson, R.C.: A general purpose algorithm for counting simple cycles and simple paths of any length (2016). arXiv preprint arXiv:1612.05531
Goel, G., Gustedt, J.: Bounded arboricity to determine the local structure of sparse graphs. In: Fomin, F.V. (ed.) WG 2006. LNCS, vol. 4271, pp. 159–167. Springer, Heidelberg (2006). doi:10.1007/11917496_15
Johnson, D.B.: Finding all the elementary circuits of a directed graph. SIAM J. Comput. 4(1), 77–84 (1975)
Kowalik, Ł.: Short cycles in planar graphs. In: Bodlaender, H.L. (ed.) WG 2003. LNCS, vol. 2880, pp. 284–296. Springer, Heidelberg (2003). doi:10.1007/978-3-540-39890-5_25
Lick, D.R., White, A.T.: \(d\)-degenerate graphs. Canad. J. Math. 22, 1082–1096 (1970)
Meeks, K.: Randomised enumeration of small witnesses using a decision oracle. In: 11th International Symposium on Parameterized, Exact Computation, IPEC 2016, Aarhus, Denmark, 24–26 August 2016, pp. 22:1–22:12 (2016)
Morrison, D.R.: Patricia-practical algorithm to retrieve information coded in alphanumeric. J. ACM 15(4), 514–534 (1968)
Papadimitriou, C.H., Yannakakis, M.: The clique problem for planar graphs. Inf. Process. Lett. 13(4), 131–133 (1981)
Richards, D.: Finding short cycles in planar graphs using separators. J. Algorithms 7(3), 382–394 (1986)
Tarjan, R.: Enumeration of the elementary circuits of a directed graph. SIAM J. Comput. 2(3), 211–216 (1973)
Uno, T., Satoh, H.: An efficient algorithm for enumerating chordless cycles and chordless paths. In: Džeroski, S., Panov, P., Kocev, D., Todorovski, L. (eds.) DS 2014. LNCS, vol. 8777, pp. 313–324. Springer, Cham (2014). doi:10.1007/978-3-319-11812-3_27
Williams, V.V., Williams, R.: Finding, minimizing, and counting weighted subgraphs. SIAM J. Comput. 42(3), 831–854 (2013)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer-Verlag GmbH Germany
About this paper
Cite this paper
Manoussakis, G. (2017). Listing All Fixed-Length Simple Cycles in Sparse Graphs in Optimal Time. In: Klasing, R., Zeitoun, M. (eds) Fundamentals of Computation Theory. FCT 2017. Lecture Notes in Computer Science(), vol 10472. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55751-8_28
Download citation
DOI: https://doi.org/10.1007/978-3-662-55751-8_28
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-55750-1
Online ISBN: 978-3-662-55751-8
eBook Packages: Computer ScienceComputer Science (R0)