DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

https://doi.org/10.7151/dmgt

Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory 22(2) (2002) 325-334
DOI: 10.7151/dmgt.1178

A CHARACTERIZATION OF ROMAN TREES

 Michael A. Henning 

School of Mathematics, Statistics, &
Information Technology
University of Natal
Private Bag X01
Pietermaritzburg, 3209 South Africa

 Abstract

A Roman dominating function (RDF) on a graph G = (V,E) is a function f: V → {0,1,2} satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. The weight of f is w(f) = ∑v ∈ V f(v). The Roman domination number is the minimum weight of an RDF in G. It is known that for every graph G, the Roman domination number of G is bounded above by twice its domination number. Graphs which have Roman domination number equal to twice their domination number are called Roman graphs. At the Ninth Quadrennial International Conference on Graph Theory, Combinatorics, Algorithms, and Applications held at Western Michigan University in June 2000, Stephen T. Hedetniemi in his principal talk entitled ``Defending the Roman Empire" posed the open problem of characterizing the Roman trees. In this paper, we give a characterization of Roman trees.

 Keywords: dominating set, Roman dominating function.

 2000 Mathematics Subject Classification: 05C069.

References

[1] E.J. Cockayne, P.A. Dreyer, S.M. Hedetniemi and S.T. Hedetniemi, Roman domination in graphs, manuscript.
[2] E.J. Cockayne, P.A. Dreyer, S.M. Hedetniemi, S.T. Hedetniemi and A. McRae, Roman domination in graphs II, manuscript.
[3] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, Inc. New York, 1998).
[4] T.W. Haynes, S.T. Hedetniemi and P.J. Slater (eds), Domination in Graphs: Advanced Topics (Marcel Dekker, Inc. New York, 1998).
[5] S.T. Hedetniemi, Defending the Roman Empire, principal talk presented at the Ninth Quadrennial International Conference on Graph Theory, Combinatorics, Algorithms, and Applications (Western Michigan University, Kalamazoo, USA, June 2000).
[6] I. Stewart, Defend the Roman Empire!, Scientific American, December 1999, 136-138, doi: 10.1038/scientificamerican1299-136.

Received 12 May 2001
Revised 25 April 2002