Efficient algorithms for Roman domination on some classes of graphs
Résumé
A 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 first answer an open question 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 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.