Counting the Number of Perfect Matchings in K 5-Free Graphs
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Counting the number of perfect matchings in graphs is a computationally hard problem. However, in the case of planar graphs, and even for K 3,3-free graphs, the number of perfect matchings can be computed efficiently. The technique to achieve this is to compute a Pfaffian orientation of a graph. In the case of K 5-free graphs, this technique will not work because some K 5-free graphs do not have a Pfaffian orientation. We circumvent this problem and show that the number of perfect matchings in K 5-free graphs can be computed in polynomial time. We also parallelize the sequential algorithm and show that the problem is in TC2. We remark that our results generalize to graphs without singly-crossing minor.
KeywordsPerfect matching Counting Complexity
We want to thank Radu Curticapean for pointing us to the literature about graphs that have no singly-crossing minor which lead to Corollary 5.11. We are greatful to Rohit Gurjar for indicating that our result on counting perfect matchings also yields the construction of a perfect matching (Corollary 5.8).
- 1.Barahona, F.: Balancing signed toroidal graphs in polynomial time. Technical report, University of Chile (1983)Google Scholar
- 2.Di Battista, G., Tamassia, R.: Incremental planarity testing. In: IEEE Symposium on Foundations of Computer Science (FOCS), pp. 436–441 (1989)Google Scholar
- 5.Curticapean, R.: Counting perfect matchings in graphs that exclude a single-crossing minor. arXiv:1406.4056 (2014)
- 7.Datta, S., Nimbhorkar, P., Thierauf, T., Wagner, F.: Isomorphism for K 3,3-free and K 5-free graphs is in log-space. In: Proceedings of the 29th Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS), pp. 145–156 (2009)Google Scholar
- 9.Elberfeld, M., Jakoby, A., Tantau, T.: Logspace versions of the theorems of Bodlaender and Courcelle. In: 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 143–152 (2010)Google Scholar
- 14.Kasteleyn, P.W.: Graph theory and crystal physics. In: Harary, F. (ed.) Graph Theory and Theoretical Physics, pp 43–110. Academic, New York (1967)Google Scholar
- 17.Little, C.H.C.: An extension of Kasteleyn’s method of enumerating the 1-factors of planar graphs. In: Holton, D.A. (ed.) Combinatorial Mathematics, volume 403 of Lecture Notes in Mathematics, pp 63–72. Springer, Berlin Heidelberg (1974)Google Scholar
- 20.Robertson, N., Seymour, P.: Excluding a graph with one crossing. In: Graph Structure Theory, pp. 669–675. American Mathematical Society (1993)Google Scholar
- 22.Thierauf, T., Wagner, F.: Reachability in K 3,3-free graphs and K 5-free graphs is in unambiguous log-space. Chic. J. Theor. Comput. Sci., To appear (2014)Google Scholar