Abstract
A k-class domination coloring (k-cd-coloring) is a partition of the vertex set of a graph G into k independent sets \(V_1,\ldots ,V_k\), where each \(V_i\) is dominated by some vertex \(u_i\) of G. The least integer k such that G admits a k-cd-coloring is called the cd-chromatic number, \(\chi _{cd}(G)\), of G. A subset S of the vertex set of a graph G is called a subclique in G if \(d_G(x,y)\ne 2\) for every \(x,y \in S\). The cardinality of a maximum subclique in G is called the subclique number, \(\omega _s(G)\), of G.
In this paper, we present algorithms to compute an optimal cd-coloring and a maximum subclique of (i) trees with time complexity O(n) and (ii) co-bipartite graphs with time complexity \(O(n^{2.5})\). This improves \(O(n^3)\) algorithms by Shalu et al. [2017, 2020]. In addition, we prove tight upper bounds for the subclique number of the class of (i) \(P_5\)-free graphs and (ii) double-split graphs.
M. A. Shalu—Supported by SERB (DST), MATRICS scheme MTR/2018/000086.
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References
Amin, S.M., Wollenberg, B.F.: Towards a smart grid. IEEE Power Energ. Mag. 3, 34–41 (2005). https://doi.org/10.1109/MPAE.2005.1507024
Androutsellis-Theotokis, S., Spinellis, D.: A survey of peer-to-peer content distribution technologies. ACM Comput. Surv. 36, 335–371 (2004). https://doi.org/10.1145/1041680.1041681
Arumugam, S., Chandrasekar, K.R., Misra, N., Philip, G., Saurabh, S.: Algorithmic aspects of dominator colorings in graphs. Comb. Algorithms 7056, 19–30 (2011). https://doi.org/10.1007/978-3-642-25011-8_2
Bocsó, D., Tuza, Z.: Dominating cliques in \({P_5}\)-free graphs. Periodica Mathematica Hungarica 21, 303–308 (1990). https://doi.org/10.1007/BF02352694
Chudnovsky, M., Robertson, N., Seymour, P., Thomas, R.: The strong perfect graph theorem. Ann. Math. 164, 51–229 (2006). https://doi.org/10.4007/annals.2006.164.51
Krithika, R., Rai, A., Saurabh, S., Tale, P.: Parameterized and exact algorithms for class domination coloring. In: Steffen, B., Baier, C., van den Brand, M., Eder, J., Hinchey, M., Margaria, T. (eds.) SOFSEM 2017. LNCS, vol. 10139, pp. 336–349. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-51963-0_26
Merouane, H.B., Haddad, M., Chellali, M., Kheddouci, H.: Dominated colorings of graphs. Graphs Comb. 31(3), 713–727 (2014). https://doi.org/10.1007/s00373-014-1407-3
Micali, S., Vazirani, V.V.: An \({O(\sqrt{|V|}|E|)}\) algorithm for finding maximum matching in general graphs. In: Proceedings of 21st IEEE Symposium on Foundations of Computer Science, pp. 17–27 (1980). https://doi.org/10.1109/SFCS.1980.12
Monti, A., Ponci, F., Benigni, A., Liu, J.: Distributed intelligence for smart grid control. In: International School on Nonsinusoidal currents and Compensation, Lagow, Poland (2010). https://doi.org/10.1109/ISNCC.2010.5524469
Shalu, M.A., Sandhya, T.P.: The cd-coloring of graphs. In: Govindarajan, S., Maheshwari, A. (eds.) CALDAM 2016. LNCS, vol. 9602, pp. 337–348. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-29221-2_29
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
Shalu, M.A., Vijayakumar, S., Sandhya, T.P.: On complexity of cd-coloring of graphs. Discret. Appl. Math. 280, 171–185 (2020). https://doi.org/10.1016/j.dam.2018.03.004
Swaminathan, V., Sundareswaran, R.: Color class domination in graphs. In: Mathematical and Experimental Physics. Narosa Publishing House (2010)
Venkatakrishnan, Y.B., Swaminathan, V.: Color class domination number of middle and central graph of \({K_{1, n}}, {C_n}, {P_n}\). Adv. Model. Optim. 12, 233–237 (2010)
West, D.B.: Introduction to Graph Theory, 2nd edn. Pearson, London (2018)
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Shalu, M.A., Kirubakaran, V.K. (2021). On cd-Coloring of Trees and Co-bipartite Graphs. In: Mudgal, A., Subramanian, C.R. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2021. Lecture Notes in Computer Science(), vol 12601. Springer, Cham. https://doi.org/10.1007/978-3-030-67899-9_16
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