Abstract
The girth of a graph is the length of its shortest cycle. Due to its relevance in graph theory, network analysis and practical fields such as distributed computing, girth-related problems have been object of attention in both past and recent literature. In this paper, we consider the problem of listing connected subgraphs with bounded girth. As a large girth is index of sparsity, this allows to extract sparse structures from the input graph. We propose two algorithms, for enumerating respectively vertex induced subgraphs and edge induced subgraphs with bounded girth, both running in O(n) amortized time per solution and using \(O(n^3)\) space. Furthermore, the algorithms can be easily adapted to relax the connectivity requirement and to deal with weighted graphs. As a byproduct, the second algorithm can be used to answer the well known question of finding the densest n-vertex graph(s) of girth k.
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Notes
- 1.
The implementation of EBG-S in the github repository: https://github.com/ikn-lab/EnumerationAlgorithms/tree/master/BoundedGirth/.
- 2.
We remark that the techniques in [6] do not extend to undirected graphs, thus motivating a separate study. In directed graphs, a u-v path and a v-u path are distinct. However, a u-v path and a v-u path may be same in undirected graphs.
References
Alon, N., Hoory, S., Linial, N.: The moore bound for irregular graphs. Gr. Comb. 18(1), 53–57 (2002)
Bollobás, B.: Extremal Graph Theory. Courier Corporation (2004)
Chandran, L.S.: A high girth graph construction. SIAM J. Discrete Math. 16(3), 366–370 (2003)
Chang, H.-C., Lu, H.-I.: Computing the girth of a planar graph in linear time. SIAM J. Comput. 42(3), 1077–1094 (2013)
Conte, A., Kanté, M.M., Otachi, Y., Uno, T., Wasa, K.: Efficient enumeration of maximal k-degenerate subgraphs in a chordal graph. In: Cao, Y., Chen, J. (eds.) COCOON 2017. LNCS, vol. 10392, pp. 150–161. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-62389-4_13
Conte, A., Kurita, K., Wasa, K., Uno, T.: Listing acyclic subgraphs and subgraphs of bounded girth in directed graphs. In: Gao, X., Du, H., Han, M. (eds.) COCOA 2017. LNCS, vol. 10628, pp. 169–181. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-71147-8_12
Ferreira, R., Grossi, R., Rizzi, R.: Output-sensitive listing of bounded-size trees in undirected graphs. In: Demetrescu, C., Halldórsson, M.M. (eds.) ESA 2011. LNCS, vol. 6942, pp. 275–286. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-23719-5_24
Itai, A., Rodeh, M.: Finding a minimum circuit in a graph. SIAM J. Comput. 7(4), 413–423 (1978)
Johnson, D.S., Yannakakis, M., Papadimitriou, C.H.: On generating all maximal independent sets. Inf. Process. Lett. 27(3), 119–123 (1988)
Kurita, K., Wasa, K., Arimura, H., Uno, T.: Efficient enumeration of dominating sets for sparse graphs. arXiv preprint arXiv:1802.07863 (2018)
Lazebnik, F., Ustimenko, V.A., Woldar, A.J.: A new series of dense graphs of high girth. Bull. Am. Math. Soc. 32(1), 73–79 (1995)
Parter, M.: Bypassing Erdős’ girth conjecture: hybrid stretch and sourcewise spanners. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds.) ICALP 2014. LNCS, vol. 8573, pp. 608–619. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-43951-7_49
Read, R.C., Tarjan, R.E.: Bounds on backtrack algorithms for listing cycles, paths, and spanning trees. Networks 3(5), 237–252 (1975)
Shioura, A., Tamura, A., Uno, T.: An optimal algorithm for scanning all spanning trees of undirected graphs. SIAM J. Comput. 26(3), 678–692 (1997)
Wasa, K., Arimura, H., Uno, T.: Efficient enumeration of induced subtrees in a K-degenerate graph. In: Ahn, H.-K., Shin, C.-S. (eds.) ISAAC 2014. LNCS, vol. 8889, pp. 94–102. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-13075-0_8
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Kurita, K., Wasa, K., Conte, A., Uno, T., Arimura, H. (2018). Efficient Enumeration of Subgraphs and Induced Subgraphs with Bounded Girth. In: Iliopoulos, C., Leong, H., Sung, WK. (eds) Combinatorial Algorithms. IWOCA 2018. Lecture Notes in Computer Science(), vol 10979. Springer, Cham. https://doi.org/10.1007/978-3-319-94667-2_17
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