Abstract
Given a graph G(V, E), a vertex subset S of G is called an open packing in G if no pair of distinct vertices in S have a common neighbour in G. The size of a largest open packing in G is called the open packing number of G and is denoted by \(\rho ^o(G)\). It is interesting to note that the open packing number is a lower bound for the total domination number in graphs with no isolated vertices [Henning and Slater, 1999]. Given a graph G and a positive integer k, Open Packing problem tests whether G has an open packing of size at least k.
It is known that Open Packing is NP-complete on split graphs (i.e., the class of \(\{2K_2,C_4,C_5\}\)-free graphs) [Ramos et al. 2014]. In this work, we complete the study on the complexity of Open Packing on H-free graphs for every graph H on at least three vertices by proving that Open Packing is (i) NP-complete on \(K_{1,3}\)-free graphs and (ii) polynomial-time solvable on \((P_4\cup rK_1)\)-free graphs for every \(r\ge 1\). Further, we prove that Open Packing is (i) NP-complete on \(K_{1,4}\)-free split graphs and (ii) polynomial-time solvable on \(K_{1,3}\)-free split graphs. We prove a similar dichotomy result on split graphs with degree restrictions on the vertices in the independent set of the clique-independent set partition of the split graph.
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References
Corneil, D.G., Perl, Y.: Clustering and domination in perfect graphs. Discret. Appl. Math. 9(1), 27–39 (1984). https://doi.org/10.1016/0166-218X(84)90088-X
Cygan, M., et al.: Parameterized Algorithms, 1st edn. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-21275-3
Damaschke, P., Muller, H., Kratsch, D.: Domination in convex and chordal bipartite graphs. Inf. Process. Lett. 31, 231–236 (1990). https://doi.org/10.1016/0020-0190(90)90147-P
Downey, R.G., Fellows, M.R.: Fixed-parameter tractability and completeness II: on completeness for W[1]. Theoret. Comput. Sci. 141(1), 109–131 (1995). https://doi.org/10.1016/0304-3975(94)00097-3
Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Springer, London (2016). https://doi.org/10.1007/978-1-4471-5559-1
Håstard, J.: Clique is hard to approximate within \(n^{1-\epsilon }\). Acta Math. 182, 105–142 (1999). https://doi.org/10.1007/BF02392825
Haynes, T.W., Hedetniemi, S., Slater, P.: Fundamentals of Domination in Graphs, 1st edn. CRC Press, Boca Raton (1998). https://doi.org/10.1201/9781482246582
Henning, M.A.: Packing in trees. Discret. Math. 186(1), 145–155 (1998). https://doi.org/10.1016/S0012-365X(97)00228-8
Henning, M.A., Slater, P.J.: Open packing in graphs. J. Comb. Math. Comb. Comput. 29, 3–16 (1999)
Henning, M.A., Yeo, A.: Total Domination in Graphs. Springer, New York (2015). https://doi.org/10.1007/978-1-4614-6525-6
Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations, pp. 85–103. Springer, Cham (1972). https://doi.org/10.1007/978-1-4684-2001-2_9
McRae, A.A.: Generalizing NP-completeness proofs for bipartite graphs and chordal graphs. Ph.D. thesis, Clemson University, USA (1995)
Rall, D.F.: Total domination in categorical products of graphs. Discussiones Mathematicae Graph Theory 25, 35–44 (2005). https://doi.org/10.7151/dmgt.1257
Ramos, I., Santos, V.F., Szwarcfiter, J.L.: Complexity aspects of the computation of the rank of a graph. Discrete Math. Theor. Comput. Sci. 16 (2014). https://doi.org/10.46298/dmtcs.2075
Renjith, P., Sadagopan, N.: The steiner tree in \( {K}_{1, r}\)-free split graphs-a dichotomy. Discret. Appl. Math. 280, 246–255 (2020). https://doi.org/10.1016/j.dam.2018.05.050
Shalu, M.A., Kirubakaran, V.K.: Total domination number and its lower bound in some subclasses of bipartite graphs. Manusript-UnderReview (2023)
Shalu, M.A., Vijayakumar, S., Sandhya, T.P.: A lower bound of the cd-chromatic number and its complexity. In: Gaur, D., Narayanaswamy, N.S. (eds.) CALDAM 2017. LNCS, vol. 10156, pp. 344–355. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-53007-9_30
West, D.B.: Introduction to Graph Theory, 2nd edn. Pearson, London (2018)
White, K., Farber, M., Pulleyblank, W.: Steiner trees, connected domination and strongly chordal graphs. Networks 15(1), 109–124 (1985). https://doi.org/10.1002/net.3230150109
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Shalu, M.A., Kirubakaran, V.K. (2024). Open Packing in H-free Graphs and Subclasses of Split Graphs. In: Kalyanasundaram, S., Maheshwari, A. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2024. Lecture Notes in Computer Science, vol 14508. Springer, Cham. https://doi.org/10.1007/978-3-031-52213-0_17
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