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Efficient algorithms for Roman domination on some classes of graphs

Abstract : 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.
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Contributor : Mathieu Liedloff Connect in order to contact the contributor
Submitted on : Tuesday, March 2, 2010 - 4:12:45 PM
Last modification on : Monday, February 7, 2022 - 9:16:01 AM

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Mathieu Liedloff, Ton Kloks, Jiping Liu, Sheng-Lung Peng. Efficient algorithms for Roman domination on some classes of graphs. Discrete Applied Mathematics, 2008, 156, pp.3400-3415. ⟨10.1016/j.dam.2008.01.011⟩. ⟨hal-00460842⟩

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