Abstract
A k-matching cover of a graph \(G\) is a union of \(k\) matchings of \(G\) which covers \(V(G)\). The matching cover number of \(G\), denoted by \(mc(G)\), is the minimum number \(k\) such that \(G\) has a \(k\)-matching cover. A matching cover of \(G\) is optimal if it consists of \(mc(G)\) matchings of \(G\). In this paper, we present an algorithm for finding an optimal matching cover of a graph on \(n\) vertices in \(O(n^3)\) time (if use a faster maximum matching algorithm, the time complexity can be reduced to \(O(nm)\), where \(m=|E(G)|\)), and give an upper bound on matching cover number of graphs. In particular, for trees, a linear-time algorithm is given, and as a by-product, the matching cover number of trees is determined.
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References
Berge, C.: Two theorems in graph theory. Proc. Nat. Acad. Sci. USA 43, 842–844 (1957)
Bondy, J.A., Murty, U.S.R.: Graph Theory. Springer, Berlin (2008)
Cunningham, W.H.: Improved bounds for matroid partition and intersection algorithms. SIAM J. Comput. 15, 948–957 (1986)
Edmonds, J.: Paths, trees, and flowers. Can. J. Math. 17, 449–467 (1965)
Gallai, T.: Über extreme Punkt- und Kantenmengen. Ann. Univ. Sci. Bp. Eötvös Sect. Math. 2, 133–138 (1959)
Gallai, T.: Maximale Systeme unabhäangiger Kanten, Magyar Tud. Akad. Mat. Kutató Int. Közl 9, 401–413 (1964)
Goldberg, A.V., Karzanov, A.V.: Maximum skew-symmetric flows. In: Algorithms—ESA ’95 (Corfu), 155–170, Lecture Notes in Computer Science, vol. 979. Springer, Berlin (1995)
Hall, P.: On representatives of subsets. J. Lond. Math. Soc. 10, 26–30 (1935)
Korte, B., Vygen, J.: Combinatorial Optimization: Theory and Algorithms, 4th edn. Springer, Berlin (2008)
Kuhn, H.W.: The Hungarian method for the assignment problem. Naval Res. Logist. Quart. 2, 83–97 (1955)
Lawler, E.: Combinatorial Optimization: Networks and Matroids. Holt, Rinehart and Winston, New York (1976)
Lovász, L., Plummer, M.D.: Matching Theory. Elsevier Science Publishers, BV, North Holland (1986)
Micali, S., Vazirani, V.V.: An \(O(V^{1/2}E)\) algorithm for finding maximum matching in general graphs. In: Proceedings of the 21st Annual IEEE Symposium on Foundations of Computer Science, pp. 17–27 (1980)
Norman, R.Z., Rabin, M.O.: An algorithm for a minimum cover of a graph. Proc. Am. Math. Soc. 10, 315–319 (1959)
Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency. Springer, Berlin (2003)
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Research supported by NSFC (11101383), NSFC (11271338), and NSFC (11201432).
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Wang, X., Song, X. & Yuan, J. On matching cover of graphs. Math. Program. 147, 499–518 (2014). https://doi.org/10.1007/s10107-013-0731-3
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DOI: https://doi.org/10.1007/s10107-013-0731-3