Efficient algorithms for Roman domination on some classes of graphs

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Abstract

A Roman dominating function of a graph G = (V, E) is a function f : V → {0, 1, 2} such that every vertex x with f (x) = 0 is adjacent to at least one vertex y with f (y) = 2. The weight of a Roman dominating function is defined to be f (V) = ∑x ∈ V f (x), and the minimum weight of a Roman dominating function on a graph G is called the Roman domination number of G. In this paper we first answer an open question mentioned in [E.J. Cockayne, P.A. Dreyer Jr., S.M. Hedetniemi, S.T. Hedetniemi, Roman domination in graphs, Discrete Math. 278 (2004) 11-22] by showing that the Roman domination number of an interval graph can be computed in linear time. We then show that the Roman domination number of a cograph (and a graph with bounded cliquewidth) can be computed in linear time. As a by-product, we give a characterization of Roman cographs. It leads to a linear-time algorithm for recognizing Roman cographs. Finally, we show that there are polynomial-time algorithms for computing the Roman domination numbers of AT-free graphs and graphs with a d-octopus. © 2008 Elsevier B.V. All rights reserved.

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Liedloff, M., Kloks, T., Liu, J., & Peng, S. L. (2008). Efficient algorithms for Roman domination on some classes of graphs. Discrete Applied Mathematics, 156(18), 3400–3415. https://doi.org/10.1016/j.dam.2008.01.011

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