article

On matching cover of graphs

Authors:

Jinjiang YuanAuthors Info & Claims

Mathematical Programming: Series A and B, Volume 147, Issue 1-2

Pages 499 - 518

https://doi.org/10.1007/s10107-013-0731-3

Published: 01 October 2014 Publication History

Abstract

A k-matching cover of a graph $$G$$ G is a union of $$k$$ k matchings of $$G$$ G which covers $$V(G)$$ V ( G ) . The matching cover number of $$G$$ G , denoted by $$mc(G)$$ m c ( G ) , is the minimum number $$k$$ k such that $$G$$ G has a $$k$$ k -matching cover. A matching cover of $$G$$ G is optimal if it consists of $$mc(G)$$ m c ( G ) matchings of $$G$$ G . In this paper, we present an algorithm for finding an optimal matching cover of a graph on $$n$$ n vertices in $$O(n^3)$$ O ( n 3 ) time (if use a faster maximum matching algorithm, the time complexity can be reduced to $$O(nm)$$ O ( n m ) , where $$m=|E(G)|$$ 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.

References

[1]

Berge, C.: Two theorems in graph theory. Proc. Nat. Acad. Sci. USA 43, 842---844 (1957)

[2]

Bondy, J.A., Murty, U.S.R.: Graph Theory. Springer, Berlin (2008)

Digital Library

[3]

Cunningham, W.H.: Improved bounds for matroid partition and intersection algorithms. SIAM J. Comput. 15, 948---957 (1986)

Digital Library

[4]

Edmonds, J.: Paths, trees, and flowers. Can. J. Math. 17, 449---467 (1965)

[5]

Gallai, T.: Über extreme Punkt- und Kantenmengen. Ann. Univ. Sci. Bp. Eötvös Sect. Math. 2, 133---138 (1959)

[6]

Gallai, T.: Maximale Systeme unabhäangiger Kanten, Magyar Tud. Akad. Mat. Kutató Int. Közl 9, 401---413 (1964)

[7]

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)

Digital Library

[8]

Hall, P.: On representatives of subsets. J. Lond. Math. Soc. 10, 26---30 (1935)

[9]

Korte, B., Vygen, J.: Combinatorial Optimization: Theory and Algorithms, 4th edn. Springer, Berlin (2008)

Digital Library

[10]

Kuhn, H.W.: The Hungarian method for the assignment problem. Naval Res. Logist. Quart. 2, 83---97 (1955)

[11]

Lawler, E.: Combinatorial Optimization: Networks and Matroids. Holt, Rinehart and Winston, New York (1976)

[12]

Lovász, L., Plummer, M.D.: Matching Theory. Elsevier Science Publishers, BV, North Holland (1986)

[13]

Micali, S., Vazirani, V.V.: An $$O(V^{1/2}E)$$O(V1/2E) 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)

Digital Library

[14]

Norman, R.Z., Rabin, M.O.: An algorithm for a minimum cover of a graph. Proc. Am. Math. Soc. 10, 315---319 (1959)

[15]

Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency. Springer, Berlin (2003)

Cited By

Ferhat DKirály ZSebő AStauffer G(2024)How many matchings cover the nodes of a graph?Mathematical Programming: Series A and B10.1007/s10107-022-01804-9203:1-2(271-284)Online publication date: 1-Jan-2024
https://dl.acm.org/doi/10.1007/s10107-022-01804-9
Dani VGupta AHayes TPettie S(2022)Wake up and join me! An energy-efficient algorithm for maximal matching in radio networksDistributed Computing10.1007/s00446-022-00426-w36:3(373-384)Online publication date: 7-May-2022
https://dl.acm.org/doi/10.1007/s00446-022-00426-w
Dani VGupta AHayes TPettie SMiller ACensor-Hillel K(2021)Brief Announcement: Wake Up and Join Me! An Energy Efficient Algorithm for Maximal Matching in Radio NetworksProceedings of the 2021 ACM Symposium on Principles of Distributed Computing10.1145/3465084.3467950(151-153)Online publication date: 21-Jul-2021
https://dl.acm.org/doi/10.1145/3465084.3467950
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Index Terms

On matching cover of graphs
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
      1. Graph algorithms

Index terms have been assigned to the content through auto-classification.

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cover image Mathematical Programming: Series A and B

Mathematical Programming: Series A and B Volume 147, Issue 1-2

October 2014

580 pages

ISSN:0025-5610

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Copyright © Copyright © 2014 Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society.

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Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 01 October 2014

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Cited By

Ferhat DKirály ZSebő AStauffer G(2024)How many matchings cover the nodes of a graph?Mathematical Programming: Series A and B10.1007/s10107-022-01804-9203:1-2(271-284)Online publication date: 1-Jan-2024
https://dl.acm.org/doi/10.1007/s10107-022-01804-9
Dani VGupta AHayes TPettie S(2022)Wake up and join me! An energy-efficient algorithm for maximal matching in radio networksDistributed Computing10.1007/s00446-022-00426-w36:3(373-384)Online publication date: 7-May-2022
https://dl.acm.org/doi/10.1007/s00446-022-00426-w
Dani VGupta AHayes TPettie SMiller ACensor-Hillel K(2021)Brief Announcement: Wake Up and Join Me! An Energy Efficient Algorithm for Maximal Matching in Radio NetworksProceedings of the 2021 ACM Symposium on Principles of Distributed Computing10.1145/3465084.3467950(151-153)Online publication date: 21-Jul-2021
https://dl.acm.org/doi/10.1145/3465084.3467950
Lang RLi TMo DShi Y(2016)A novel method for analyzing inverse problem of topological indices of graphs using competitive agglomerationApplied Mathematics and Computation10.1016/j.amc.2016.06.048291:C(115-121)Online publication date: 1-Dec-2016
https://dl.acm.org/doi/10.1016/j.amc.2016.06.048
Soltan SYannakakis MZussman G(2015)Joint Cyber and Physical Attacks on Power GridsACM SIGMETRICS Performance Evaluation Review10.1145/2796314.274584643:1(361-374)Online publication date: 15-Jun-2015
https://dl.acm.org/doi/10.1145/2796314.2745846
Soltan SYannakakis MZussman GLin BXu JSengupta SShah D(2015)Joint Cyber and Physical Attacks on Power GridsProceedings of the 2015 ACM SIGMETRICS International Conference on Measurement and Modeling of Computer Systems10.1145/2745844.2745846(361-374)Online publication date: 15-Jun-2015
https://dl.acm.org/doi/10.1145/2745844.2745846

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