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On cd-Coloring of Trees and Co-bipartite Graphs

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Algorithms and Discrete Applied Mathematics (CALDAM 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12601))

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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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Correspondence to V. K. Kirubakaran .

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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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  • DOI: https://doi.org/10.1007/978-3-030-67899-9_16

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