Abstract
A vertex set partition of a graph G into k independent sets \(V_1, V_2, \ldots , V_k\) is called a k-color domination partition (k-cd-coloring) of G if there exists a vertex \(u_i\in V(G)\) such that \(u_i\) dominates \(V_i\) in G for \(1 \le i \le k\). We prove that deciding whether a graph G admits a k-cd-coloring is in P for \(k\le 3\) and NP-complete for \(k>3\). We also characterize the class of all 3-cd-colorable graphs. In addition, we provide a polynomial time algorithm to find an optimal cd-coloring of \(P_4\)-free graphs and split graphs.
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Notes
- 1.
Due to space constraints, proofs of the results marked with a [\(\varvec{*} \)] is deferred to a longer version of the paper.
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The authors wish to thank the anonymous referees whose suggestions improved the presentation of this paper.
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Shalu, M.A., Sandhya, T.P. (2016). The cd-Coloring of Graphs. In: Govindarajan, S., Maheshwari, A. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2016. Lecture Notes in Computer Science(), vol 9602. Springer, Cham. https://doi.org/10.1007/978-3-319-29221-2_29
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DOI: https://doi.org/10.1007/978-3-319-29221-2_29
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