Abstract
Let G be a graph on n vertices with bounded treewidth. We use \(f_k(G)\) to denote the number of spanning forests of G with k components. Given a tree decomposition of width at most p of G, we present an algorithm to compute \(f_k(G)\) for \(k = 1,2, \cdots , n\). The running time of our algorithm is \(O((B(p+1))^3pn^3)\), where \(B(p+1)\) is the \((p+1)\)-th Bell number.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Benjamini, I., Lyons, R., Peres, Y., Schramm, O.: Uniform spanning forests. Ann. Probab. 29, 1–65 (2001)
Brown, T.J., Mallion, R.B., Pollak, P., Roth, A.: Some methods for counting the spanning trees in labelled molecular graphs, examined in relation to certain fullerenes. Discrete Appl. Math. 67, 51–66 (1996)
Caracciolo, S., Jacobsen, J.L., Saleur, H., Sokal, A.D., Sportiello, A.: Fermionic field theory for trees and forests. Phys. Rev. Lett. 93, 080601 (2004)
Deng, Y., Garoni, T.M., Sokal, A.D.: Ferromagnetic phase transition for the spanning-forest model (\(q\rightarrow 0\) limit of the Potts model) in three or more dimensions. Phys. Rev. Lett. 98, 030602 (2007)
Liu, C.J., Chow, Y.: Enumeration of forests in a graph. Proc. Am. Math. Soc. 83, 659–662 (1981)
Stark, D.: The asymptotic number of spanning forests of complete bipartite labelled graphs. Discrete Math. 313, 1256–1261 (2013)
Myrvold, W.: Counting \(k\)-component forests of a graph. Networks 22, 647–652 (1992)
Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: Computing the Tutte polynomial in vertex-exponential time. In: Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, pp. 677-686. IEEE. New York (2008)
Teranishi, Y.: The number of spanning forests of a graph. Discrete Math. 290, 259–267 (2005)
Jaeger, F., Vertigan, D., Welsh, D.: On the computational complexity of the Jones and Tutte polynomials. Math. Proc. Camb. Philos. Soc. 108, 35–53 (1990)
Vertigan, D.L., Welsh, D.J.A.: The computational complexity of the Tutte plane: the bipartite case. Combin. Probab. Comput. 1, 181–187 (1992)
Gebauer, H., Okamoto, Y.: Fast exponential-time algorithms for the forest counting and the Tutte polynomial computation in graph classes. Int. J. Found. Comput. Sci. 20, 25–44 (2009)
Kirchhoff, G.: Ueber die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme geführt wird. Ann. Phys. Chem. 72, 497–508 (1847)
Palmer, E.M.: On the spanning tree packing number of a graph: a survey. Discrete Math. 230, 13–21 (2001)
Comellas, F., Miralles, A., Liu, H., Zhang, Z.: The number of spanning trees of an infinite family of outerplanar, small-world and self-similar graphs. Phys. A 392, 2803–2806 (2013)
Bogdanowicz, Z.R.: Formulas for the number of spanning trees in a fan. Appl. Math. Sci. 2, 781–786 (2008)
Xiao, Y., Zhao, H.: New method for counting the number of spanning trees in a two-tree network. Phys. A 392, 4576–4583 (2013)
Zhang, Z., Wu, B., Comellas, F.: The number of spanning trees in Apollonian networks. Discrete Appl. Math. 169, 206–213 (2014)
Stones, R.J.: Computing the number of \(h\)-edge spanning forests in complete bipartite graphs. Discrete Math. Theor. Comput. Sci. 16, 313–326 (2014)
Robertson, N., Seymour, P.D.: Graph minors. II. Algorithmic aspects of tree-width. J. Algorithms 7, 309–322 (1986)
Courcelle, B.: The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Inf. Comput. 85, 12–75 (1990)
Arnborg, S., Lagergren, J., Seese, D.: Easy problems for tree-decomposable graphs. J. Algorithms 12, 308–340 (1991)
Borie, R.B., Parker, R.G., Tovey, C.A.: Automatic generation of linear-time algorithms from predicate calculus descriptions of problems on recursively constructed graph families. Algorithmica 7, 555–581 (1992)
Bodlaender, H.L., Cygan, M., Kratsch, S., Nederlof, J.: Deterministic single exponential time algorithms for connectivity problems parameterized by treewidth. Inf. Comput. 243, 86–111 (2015)
Berend, D., Tassa, T.: Improved bounds on Bell numbers and on moments of sums of random variables. Probab. Math. Stat. 30, 185–205 (2010)
Bodlaender, H.L.: A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25, 1305–1317 (1996)
Bodlaender, H.L., Koster, A.M.: Treewidth computations I. Upper bounds. Inf. Comput. 208, 259–275 (2010)
Bodlaender, H.L.: A partial \(k\)-arboretum of graphs with bounded treewidth. Theor. Comput. Sci. 209, 1–45 (1998)
Bondy, J.A., Murty, U.S.R.: Graph Theory. Springer, London (2008)
Kloks, T.: Treewidth: Computations and Approximations. Lecture Notes in Computer Science, vol. 842. Springer, Berlin (1994)
Cygan, M., Nederlof, J., Pilipczuk, M., Pilipczuk, M., van Rooij, J.M.M., Wojtaszczyk, J.O.: Solving connectivity problems parameterized by treewidth in single exponential time. In: Proceedings of Foundations of Computer Science, pp. 150–159. IEEE (2011)
Acknowledgements
We greatly appreciate the anonymous referees for their comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was supported by the National Natural Science Foundation of China (Nos. 11571135, 11671320 and 11601430).
Rights and permissions
About this article
Cite this article
Wan, PF., Chen, XZ. Computing the Number of k-Component Spanning Forests of a Graph with Bounded Treewidth. J. Oper. Res. Soc. China 7, 385–394 (2019). https://doi.org/10.1007/s40305-019-00241-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40305-019-00241-4