International audienceA Roman dominating function of a graph $G=(V,E)$ is a function $f:V \rightarrow \{0,1,2\}$ 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 $f(V)=\sum_{x\in V}f(x)$, 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 answer an open problem mentioned in [E. J. Cockayne, P. A. Jr. Dreyer, S. M. Hedetniemi, S. T. Hedetniemi, Roman domination in graphs, Discrete Math. 278, (2004), pp. 11--22] by showing that the Roman domination number of an interval graph can be computed in linear time. We also show that the Roman domination number of a cograph can be computed in linear time. Besides, we show that there are polynomial time algorithms for computing the Roman domination numbers of AT-free graphs and graphs with a $d$-octopus
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.