Abstract
A bipartite graph G=(U,W,E) with vertex set V=U∪W is convex if there exists an ordering of the vertices of W such that for each u∈U, the neighbors of u are consecutive in W. A compact representation of a convex bipartite graph for specifying such an ordering can be computed in O(|V|+|E|) time. The paired-domination problem on bipartite graphs has been shown to be NP-complete. The complexity of the paired-domination problem on convex bipartite graphs has remained unknown. In this paper, we present an O(|V|) time algorithm to solve the paired-domination problem on convex bipartite graphs given a compact representation. As a byproduct, we show that our algorithm can be directly applied to solve the total domination problem on convex bipartite graphs in the same time bound.
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00224-011-9378-8/MediaObjects/224_2011_9378_Fig1_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00224-011-9378-8/MediaObjects/224_2011_9378_Fig2_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00224-011-9378-8/MediaObjects/224_2011_9378_Fig3_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00224-011-9378-8/MediaObjects/224_2011_9378_Fig4_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00224-011-9378-8/MediaObjects/224_2011_9378_Fig5_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00224-011-9378-8/MediaObjects/224_2011_9378_Fig6_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00224-011-9378-8/MediaObjects/224_2011_9378_Fig7_HTML.gif)
Similar content being viewed by others
References
Arikati, S.R., Pandu Rangan, C.: Linear algorithm for optimal path cover problem on interval graphs. Inf. Process. Lett. 35, 149–153 (1990)
Asdre, K., Nikolopoulos, S.D.: NP-completeness results for some problems on subclasses of bipartite and chordal graphs. Theor. Comput. Sci. 381, 248–259 (2007)
Bang-Jensen, J., Huang, J., MacGillivray, G., Yeo, A.: Domination in convex bipartite and convex-round graphs. Tech. Rep. PP-1999-08, University of Southern Denmark (1999)
Booth, K., Lueker, G.: Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms. J. Comput. Syst. Sci. 13, 335–379 (1976)
Bose, P., Chan, A., Dehne, F., Latzel, M.: Coarse grained parallel maximum matching in convex bipartite graphs. In: 13th International Parallel Processing Symposium (IPPS’99), pp. 125–129 (1999)
Brodal, G.S., Georgiadis, L., Hansen, K.A., Katriel, I.: Dynamic matchings in convex bipartite graphs. Lect. Notes Comput. Sci. 4708, 406–417 (2007)
Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes: A Survey. Society for Industrial and Applied Mathematics, Philadelphia (1999)
Brandstädt, A., Eschen, E.M., Sritharan, R.: The induced matching and chain subgraph cover problems for convex bipartite graphs. Theor. Comput. Sci. 381, 260–265 (2007)
Chan, A., Dehne, F., Bose, P., Latzel, M.: Coarse grained parallel algorithms for graph matching. Parallel Comput. 34, 47–62 (2008)
Chang, M.S., Peng, S.L., Liaw, J.L.: Deferred-query: an efficient approach for some problems on interval graphs. Networks 34, 1–10 (1999)
Chen, L., Lu, C., Zeng, Z.: A linear-time algorithm for paired-domination problem in strongly chordal graphs. Inf. Process. Lett. 110, 20–23 (2009)
Chen, L., Lu, C., Zeng, Z.: Labelling algorithms for paired-domination problems in block and interval graphs. J. Comb. Optim. 19, 457–470 (2010)
Cheng, T.C.E., Kang, L., Ng, C.T.: Paired domination on interval and circular-arc graphs. Discrete Appl. Math. 155, 2077–2086 (2007)
Cheng, T.C.E., Kang, L., Shan, E.: A polynomial-time algorithm for the paired-domination problem on permutation graphs. Discrete Appl. Math. 157, 262–271 (2009)
Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 3rd edn. MIT Press, Cambridge (2009)
Damaschke, P., Müller, H., Kratsch, D.: Domination in convex and chordal bipartite graphs. Inf. Process. Lett. 36, 231–236 (1990)
Dekel, E., Sahni, S.: A parallel matching for convex bipartite graphs and applications to scheduling. J. Parallel Distrib. Comput. 1, 185–205 (1984)
Gallo, G.: An O(nlogn) algorithm for the convex bipartite matching problem. Oper. Res. Lett. 3, 31–34 (1984)
Glover, F.: Maximum matching in a convex bipartite graph. Nav. Res. Logist. Q. 14, 313–316 (1967)
Haynes, T.W., Slater, P.J.: Paired-domination in graphs. Networks 32, 199–206 (1998)
Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Fundamentals of Domination in Graphs. Dekker, New York (1998)
Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Domination in Graphs: Advanced Topics. Dekker, New York (1998)
Hung, R.W., Chang, M.S.: Linear-time certifying algorithms for the path cover and Hamiltonian cycle problems on interval graphs. Appl. Math. Lett. 24, 648–652 (2011)
Hung, R.W., Laio, C.H., Wang, C.K.: Efficient algorithm for the paired-domination problem in convex bipartite graphs. In: Proceedings of the International MultiConference of Engineers and Computer Scientists 2010 (IMECS’2010), vol. I, pp. 365–369 (2010)
Katriel, I.: Matchings in node-weighted convex bipartite graphs. INFORMS J. Comput. 20, 205–211 (2008)
Korpelainen, N.: A polynomial-time algorithm for the dominating induced matching problem in the class of convex graphs. Electron. Notes Discrete Math. 32, 133–140 (2009)
Lappas, E., Nikolopoulos, S.D., Palios, L.: An O(n)-time algorithm for the paired-domination problem on permutation graphs. Lect. Notes Comput. Sci. 5874, 368–379 (2009)
Liang, Y.D., Chang, M.S.: Minimum feedback vertex sets in cocomparability graphs and convex bipartite graphs. Acta Inform. 34, 337–346 (1997)
Lipski, W., Preparata, F.: Efficient algorithms for finding maximum matchings in convex bipartite graphs and related problems. Acta Inform. 15, 329–346 (1981)
Müller, H.: Hamiltonian circuits in chordal bipartite graphs. Discrete Math. 156, 291–298 (1996)
Nussbaum, D., Pu, S., Sack, J.R., Uno, T., Zarrabi-Zadeh, H.: Finding maximum edge bicliques in convex bipartite graphs. Lect. Notes Comput. Sci. 6196, 140–149 (2010)
Park, E., Park, K.: An improved Boolean circuit for maximum matching in a convex bipartite graph. Fundam. Inform. 84, 81–107 (2008)
Qiao, H., Kang, L., Cardei, M., Du, D.Z.: Paired-domination of trees. J. Glob. Optim. 25, 43–54 (2003)
Soares, J., Stefanes, M.A.: Algorithms for maximum independent set in convex bipartite graphs. Algorithmica 53, 35–49 (2009)
Srinivasan, A., Madhukar, K., Nagavamsi, P., Pandu Rangan, C., Chang, M.S.: Edge domination on bipartite permutation graphs and cotriangulated graphs. Inf. Process. Lett. 56, 165–171 (1995)
Steiner, G., Yeoman, J.: A linear time algorithm for maximum matchings in convex, bipartite graphs. Comput. Math. Appl. 31(12), 91–96 (1996)
Yang, S.J.: Efficient algorithms to solve the link-orientation problem for multi-square, convex-bipartite, and convex-split networks. Inf. Sci. 171, 475–493 (2005)
Yen, W.C.K.: The bottleneck independent domination on the classes of bipartite graphs and block graphs. Inf. Sci. 157, 199–215 (2003)
Yu, C.W., Chen, G.H., Ma, T.H.: On the complexity of the k-chain subgraph cover problem. Theor. Comput. Sci. 205, 85–98 (1998)
Acknowledgements
The author gratefully acknowledges the helpful comments and suggestions of the reviewers, which have improved the presentation. We also deeply appreciate the anonymous reviewers for giving the comment that the proposed approach can be applied to the total domination problem in convex bipartite graphs.
Author information
Authors and Affiliations
Corresponding author
Additional information
A preliminary version of this paper has appeared in: Proceedings of the International MultiConference of Engineers and Computer Scientists 2010 (IMECS’2010), Hong Kong, vol. I, pp. 365–369 (2010) [24].
Rights and permissions
About this article
Cite this article
Hung, RW. Linear-Time Algorithm for the Paired-Domination Problem in Convex Bipartite Graphs. Theory Comput Syst 50, 721–738 (2012). https://doi.org/10.1007/s00224-011-9378-8
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00224-011-9378-8