Maximum weight induced matching in some subclasses of bipartite graphs

Abstract

A subset \(M\subseteq E\) of edges of a graph \(G=(V,E)\) is called a matching in G if no two edges in M share a common vertex. A matching M in G is called an induced matching if G[M], the subgraph of G induced by M, is the same as G[S], the subgraph of G induced by \(S=\{v \in V |\)v is incident on an edge of \(M\}\). The Maximum Induced Matching problem is to find an induced matching of maximum cardinality. Given a graph G and a positive integer k, the Induced Matching Decision problem is to decide whether G has an induced matching of cardinality at least k. The Maximum Weight Induced Matching problem in a weighted graph \(G=(V,E)\) in which the weight of each edge is a positive real number, is to find an induced matching such that the sum of the weights of its edges is maximum. It is known that the Induced Matching Decision problem and hence the Maximum Weight Induced Matching problem is known to be NP-complete for general graphs and bipartite graphs. In this paper, we strengthened this result by showing that the Induced Matching Decision problem is NP-complete for star-convex bipartite graphs, comb-convex bipartite graphs, and perfect elimination bipartite graphs, the subclasses of the class of bipartite graphs. On the positive side, we propose polynomial time algorithms for the Maximum Weight Induced Matching problem for circular-convex bipartite graphs and triad-convex bipartite graphs by making polynomial time reductions from the Maximum Weight Induced Matching problem in these graph classes to the Maximum Weight Induced Matching problem in convex bipartite graphs.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

References

  1. Bao FS, Zhang Y (2012) A review of tree convex sets test. Comput Intell 28:358–372

    MathSciNet  Article  Google Scholar 

  2. Cameron K (1989) Induced matchings. Discrete Appl Math 24:97–102

    MathSciNet  Article  Google Scholar 

  3. Cameron K, Sritharan R, Tang Y (2003) Finding a maximum induced matching in weakly chordal graphs. Discrete Math 266:133–142

    MathSciNet  Article  Google Scholar 

  4. Chen H, Lei Z, Liu T, Tang Z, Wang C, Xu K (2016) Complexity of domination, hamiltonicity and treewidth for tree convex bipartite graphs. J Comb Optim 32:95–110

    MathSciNet  Article  Google Scholar 

  5. Duckworth W, Manlove DF, Zito M (2005) On the approximability of the maximum induced matching problem. J Discrete Algorithms 3:79–91

    MathSciNet  Article  Google Scholar 

  6. Golumbic MC, Gauss CF (1978) Perfect elimination and chordal bipartite graphs. J Graph Theory 2:155–163

    MathSciNet  Article  Google Scholar 

  7. Golumbic MC, Lewenstein M (2000) New results on induced matchings. Discrete Appl Math 101:157–165

    MathSciNet  Article  Google Scholar 

  8. Jiang W, Liu T, Wang C, Xu K (2013) Feedback vertex sets on restricted bipartite graphs. Theor Comput Sci 507:41–51

    MathSciNet  Article  Google Scholar 

  9. Klemz B, Rote G (2017) Linear-time algorithms for maximum-weight induced matchings and minimum chain covers in convex bipartite graphs. arXiv:1711.04496

  10. Kobler D, Rotics U (2003) Finding maximum induced matchings in subclasses of claw-free and P5-free graphs, and in graphs with matching and induced matching of equal maximum size. Algorithmica 37:327–346

    MathSciNet  Article  Google Scholar 

  11. Liang YD, Blum N (1995) Circular convex bipartite graphs: maximum matching and hamiltonial circuits. Inf Process Lett 56:215–219

    Article  Google Scholar 

  12. Liu T (2014) Restricted bipartite graphs: comparison and hardness results. In: Algorithmic aspects in information and management, Lecture notes in computer science, vol 8546, pp 241–252

  13. Liu T, Lu Z, Xu K (2015) Tractable connected domination for restricted bipartite graphs. J Comb Optim 29:247–256

    MathSciNet  Article  Google Scholar 

  14. Liu T, Lu M, Lu Z, Xu K (2014) Circular convex bipartite graphs: feedback vertex sets. Theor Comput Sci 556:55–62

    MathSciNet  Article  Google Scholar 

  15. Lozin VV (2002) On maximum induced matchings in bipartite graphs. Inf Process Lett 81:7–11

    MathSciNet  Article  Google Scholar 

  16. Pandey A, Panda BS (2019) Domination in some subclasses of bipartite graphs. Discrete Appl Math 252:51–66

    MathSciNet  Article  Google Scholar 

  17. Pandey A, Panda BS, Dane P, Kashyap M (2017) Induced matching in some subclasses of bipartite graphs. In: Conference on algorithms and discrete applied mathematics, pp 308–319. Springer

  18. Song Y, Liu T, Xu K (2012) Independent domination on tree-convex bipartite graphs. In: FAW-AAIM, Lecture notes in computer science, vol 7285, pp 129–138

  19. Stockmeyer LJ, Vazirani VV (1982) NP-completeness of some generalizations of the maximum matching problem. Inf Process Lett 15:14–19

    MathSciNet  Article  Google Scholar 

  20. Wang C, Chen H, Lei Z, Tang Z, Liu T, Xu K (2014) Tree Convex BipartiteGraphs: NP-complete domination, hamiltonicity and treewidth. In: FAW 2014, Lecture notes in computer science, vol 8947, pp 252–263

  21. Zito M (1999) Maximum induced matchings in regular graphs and trees. In: WG’99: 25th international workshop on graph-theoretic concepts in computer science, Lecture notes in computer science, vol 1665, pp 89–100

Download references

Acknowledgements

The first author thanks the SERB, Department of Science and Technology for their support vide Diary No. SERB/F/12949/2018-2019. The third author wants to thank the Department of Science and Technology (INSPIRE) for their support.

Author information

Affiliations

Authors

Corresponding author

Correspondence to B. S. Panda.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Panda, B.S., Pandey, A., Chaudhary, J. et al. Maximum weight induced matching in some subclasses of bipartite graphs. J Comb Optim (2020). https://doi.org/10.1007/s10878-020-00611-2

Download citation

Keywords

  • Matching
  • Induced matching
  • Bipartite graphs
  • Graph algorithm
  • NP-complete