Elsevier

Discrete Applied Mathematics

Volume 156, Issue 18, 28 November 2008, Pages 3400-3415
Discrete Applied Mathematics

Efficient algorithms for Roman domination on some classes of graphs

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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)=xVf(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.

Keywords

Graph algorithms
Roman domination
Interval graphs
Cographs
AT-free graphs

Cited by (0)

The second and the third authors were partially supported by NSERC of Canada. The last author was supported by the National Science Council of Taiwan under grant NSC 95-2221-E-259-010.

1

Currently, this author is visiting the School of Computing, University of Leeds, Leeds LS2 9JT, UK.

2

During the redaction of this paper the sad news reached us that Jiping Liu had passed away.