Abstract
We investigate the computational complexity of finding a minimum Roman k-dominating function (RKDF) on split graphs. We prove that RKDF on split graphs is NP-complete on \(K_{1,2k+3}\)-free split graphs. We also show that finding RKDF on star-convex bipartite graphs and comb-convex bipartite graphs are NP-complete. Further, we also show that finding RKDF on bipartite chain graphs is polynomial-time solvable, which is a non-trivial subclass of comb-convex bipartite graphs. On the parameterized front, we show that finding RKDF on split graphs is in W[1]-hard when the parameter is the solution size. From the approximation perspective, we show that there is no constant factor approximation algorithm for RKDF.
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Mohanapriya, A., Renjith, P., Sadagopan, N. (2023). Roman k-Domination: Hardness, Approximation and Parameterized Results. In: Lin, CC., Lin, B.M.T., Liotta, G. (eds) WALCOM: Algorithms and Computation. WALCOM 2023. Lecture Notes in Computer Science, vol 13973. Springer, Cham. https://doi.org/10.1007/978-3-031-27051-2_29
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