Efficient algorithms for Roman domination on some classes of graphs

  • Authors:

  • Mathieu Liedloff;Ton Kloks;Jiping Liu;Sheng-Lung Peng

  • Affiliations:

  • Université Paul Verlaine-Metz, LITA, 57045 Metz Cedex 01, France;-;Department of Mathematics and Computer Science, The University of Lethbridge, Alberta, T1K 3M4, Canada;Department of Computer Science and Information Engineering, National Dong Hwa University, Hualien 974, Taiwan

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2008

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Abstract

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"@?"Vf(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.