Efficient algorithms for Roman domination on some classes of graphs

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.