A Roman dominating function of a graph is a function such that every vertex with is adjacent to at least one vertex with . The weight of a Roman dominating function is defined to be , and the minimum weight of a Roman dominating function on a graph is called the Roman domination number of . 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 -free graphs and graphs with a -octopus.
The second and the third authors were partially supported by NSERC of Canada. The last author was supported by the National Science Council of Taiwan under grant NSC 95-2221-E-259-010.