Abstract
A paired-dominating set of a graph G is a dominating set S of G such that the subgraph of G induced by S has a perfect matching. Haynes and Slater (Networks 32(3):199–206, 1998) introduced the concept of paired-domination and showed that the problem of determining minimum paired-dominating sets is NP-complete on general graphs. Ever since then many algorithmic results are studied on some important classes of graphs. In this paper, we extend the results by providing an \(O(n^2)\)-time algorithm on distance-hereditary graphs.
Similar content being viewed by others
References
Alvarado, J.D., Dantas, S., Rautenbach, D.: Perfectly relating the domination, total domination, and paired domination numbers of a graph. Discrete Math. 338(8), 1424–1431 (2015)
Arnborg, S., Proskurowski, A.: Linear time algorithms for NP-hard problems restricted to partial \(k\)-trees. Discrete Appl. Math. 23(1), 11–24 (1989)
Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., Protasi, M.: Complexity and Approximation. Springer, Berlin (1999)
Bandelt, H.-J., Mulder, H.M.: Distance-hereditary graphs. J. Combin. Theory Ser. B 41(2), 182–208 (1986)
Bertossi, A.A.: Dominating sets for split and bipartite graphs. Inf. Process. Lett. 19(1), 37–40 (1984)
Beyer, T., Proskurowski, A., Hedetniemi, S., Mitchell, S.: Independent domination in trees. In: Proceedings of SEICCGTC’77, pp. 321–328. Utilitas Math., Winnipeg, Man. (1977)
Booth, K.S., Johnson, J.H.: Dominating sets in chordal graphs. SIAM J. Comput. 11(1), 191–199 (1982)
Brandstädt, A., Kratsch, D.: On the restriction of some NP-complete graph problems to permutation graphs. In: the Proceedings of FCT’85, Volume 199 of LNCS, pp. 53–62. Springer, Berlin (1985)
Brandstädt, A., Mosca, R.: Weighted efficient domination for \(P_5\)-free and \(P_6\)-free graphs. SIAM J. Discrete Math. 30(4), 2288–2303 (2016)
Chang, G.J.: Labeling algorithms for domination problems in sun-free chordal graphs. Discrete Appl. Math. 22(1), 21–34 (1988)
Chang, G.J.: Algorithmic aspects of domination in graphs. In: Handbook of Combinatorial Optimization, Vol. 3, pp. 339–405. Kluwer Acad. Publ., Boston, MA (1998)
Chang, M.-S.: Efficient algorithms for the domination problems on interval and circular-arc graphs. SIAM J. Comput. 27(6), 1671–1694 (1998)
Chang, M.-S., Hsieh, S.-Y., Chen, G.-H.: Dynamic programming on distance-hereditary graphs. In: the Proceedings of ISAAC’97, Volume 1350 of LNCS, pp. 344–353. Springer, Berlin (1997)
Chang, M.-S., Wu, S.-C., Chang, G.J., Yeh, H.-G.: Domination in distance-hereditary graphs. Discrete Appl. Math. 116(1–2), 103–113 (2002)
Chao, H.S., Hsu, F.R., Lee, R.C.T.: An optimal algorithm for finding the minimum cardinality dominating set on permutation graphs. Discrete Appl. Math. 102(3), 159–173 (2000)
Chen, L., Lu, C., Zeng, Z.: Hardness results and approximation algorithms for (weighted) paired-domination graphs. Theor. Comput. Sci. 410(47–49), 5063–5071 (2009)
Chen, L., Lu, C., Zeng, Z.: A linear-time algorithm for paired-domination problem in strongly chordal graphs. Inf. Process. Lett. 110(1), 20–23 (2009)
Chen, L., Lu, C., Zeng, Z.: Labelling algorithms for paired-domination problems in block and interval graphs. J. Comb. Optim. 19(4), 457–470 (2010)
Cheng, T.C.E., Kang, L., Shan, E.: A polynomial-time algorithm for the paired-domination problem on permutation graphs. Discrete Appl. Math. 157(2), 262–271 (2009)
Cockayne, E., Goodman, S., Hedetniemi, S.: A linear algorithm for the domination number of a tree. Inf. Process. Lett. 4(2), 41–44 (1975)
Colbourn, C.J., Stewart, L.K.: Permutation graphs: connected domination and Steiner trees. Discrete Math. 86(1–3), 179–189 (1990)
Corneil, D.G., Perl, Y.: Clustering and domination in perfect graphs. Discrete Appl. Math. 9, 27–39 (1984)
Corneil, D.G., Stewart, L.K.: Dominating sets in perfect graphs. Discrete Math. 86(1–3), 145–164 (1990)
D’Atri, A., Moscarini, M.: Distance-hereditary graphs, Steiner trees, and connected domination. SIAM J. Comput. 17(3), 521–538 (1988)
Desormeaux, W.J., Henning, M.A.: Paired domination in graphs: a survey and recent results. Util. Math. 94, 101–166 (2014)
Farber, M.: Independent domination in chordal graphs. Oper. Res. Lett. 1(4), 134–138 (1982)
Farber, M.: Domination, independent domination, and duality in strongly chordal graphs. Discrete Appl. Math. 7(2), 115–130 (1984)
Farber, M., Keil, J.M.: Domination in permutation graphs. J. Algorithms 6(3), 309–321 (1985)
Foucaud, F., Mertzios, G.B., Naserasr, R., Parreau, A., Valicov, P.: Identification, location-domination and metric dimension on interval and permutation graphs. II. Algorithms and complexity. Algorithmica 78(3), 914–944 (2017)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1979)
Giannopoulou, A.C., Mertzios, G.B.: New geometric representations and domination problems on tolerance and multitolerance graphs. SIAM J. Discrete Math. 30(3), 1685–1725 (2016)
Hammer, P.L., Maffray, F.: Completely separable graphs. Discrete Appl. Math. 27(1–2), 85–99 (1990)
Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Domination in Graphs: Advanced Topics. Marcel Dekker, New York (1998)
Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Fundamentals of Domination in Graphs. Marcel Dekker, New York (1998)
Haynes, T.W., Slater, P.J.: Paired-domination in graphs. Networks 32(3), 199–206 (1998)
Hedetniemi, S.T., Laskar, R.C.: Bibliography on domination in graphs and some basic definitions of domination parameters. Discrete Math. 86(1–3), 257–277 (1990)
Hedetniemi, S.T., Laskar, R.C.: Topics on Domination. North-Holland Publishing, Amsterdam (1991)
Henning, M.A., Pradhan, D.: Algorithmic aspects of upper paired-domination in graphs. Theor. Comput. Sci. 804, 98–114 (2020)
Hochbaum, D.S.: Approximation Algorithms for NP-Hard Problems. PWS Publishing, New Orleans (1996)
Howorka, E.: A characterization of distance-hereditary graphs. Q. J. Math. Oxford Ser. (2) 28(112), 417–420 (1977)
Hsieh, S.-Y., Ho, C.-W., Hsu, T.-S., Ko, M.-T., Chen, G.-H.: Characterization of efficiently parallel solvable problems on distance-hereditary graphs. SIAM J. Discrete Math. 15(4), 488–518 (2002)
Hsu, W.L., Tsai, K.-H.: Linear time algorithms on circular-arc graphs. Inf. Process. Lett. 40(3), 123–129 (1991)
Kang, L.: Variations of dominating set problem. In: Pardalos, P., Du, D.Z., Graham, R. (eds.) Handbook of Combinatorial Optimization, 2nd edn, pp. 3363–3394. Springer, Berlin (2013)
Kao, M.-J., Chen, H.-L., Lee, D.T.: Capacitated domination: problem complexity and approximation algorithms. Algorithmica 72(1), 1–43 (2015)
Keil, J.M.: Total domination in interval graphs. Inf. Process. Lett. 22(4), 171–174 (1986)
Keil, J.M.: The complexity of domination problems in circle graphs. Discrete Appl. Math. 42(1), 51–63 (1993)
Lappas, E., Nikolopoulos, S.D., Palios, L.: An \(O(n)\)-time algorithm for the paired domination problem on permutation graphs. Eur. J. Combin. 34(3), 593–608 (2013)
Laskar, R., Pfaff, J.: Domination and irredundance in split graphs. Technical Report 430, Clemson University, Dept. Math. Sciences (1983)
Laskar, R., Pfaff, J., Hedetniemi, S.M., Hedetniemi, S.T.: On the algorithmic complexity of total domination. SIAM J. Algebr. Discrete Methods 5(3), 420–425 (1984)
Lin, C.-C., Tu, H.-L.: A linear-time algorithm for paired-domination on circular-arc graphs. Theor. Comput. Sci. 591, 99–105 (2015)
Lu, C., Wang, B., Wang, K., Wu, Y.: Paired-domination in claw-free graphs with minimum degree at least three. Discrete Appl. Math. 257, 250–259 (2019)
Pfaff, J., Laskar, R., Hedetniemi, S.T.: NP-completeness of total and connected domination and irredundance for bipartite graphs. Technical Report 428, Clemson University, Dept. Math. Sciences (1983)
Qiao, H., Kang, L., Cardei, M., Du, D.-Z.: Paired-domination of trees. J. Global Optim. 25(1), 43–54 (2003)
White, K., Farber, M., Pulleyblank, W.: Steiner trees, connected domination and strongly chordal graphs. Networks 15(1), 109–124 (1985)
Yeh, H.-G., Chang, G.J.: Weighted connected domination and Steiner trees in distance-hereditary graphs. Discrete Appl. Math. 87(1–3), 245–253 (1998)
Acknowledgements
This work is partially supported by the Ministry of Science and Technology under the Grants Nos. MOST 106-2221-E-019-014, and MOST 107-2221-E-019-016.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Lin, CC., Ku, KC. & Hsu, CH. Paired-Domination Problem on Distance-Hereditary Graphs. Algorithmica 82, 2809–2840 (2020). https://doi.org/10.1007/s00453-020-00705-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00453-020-00705-7