Discussiones Mathematicae Graph Theory 33(2) (2013) 337-346
doi: 10.7151/dmgt.1669

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Strong equality between the Roman domination and independent Roman domination numbers in trees

Mustapha Chellali

LAMDA-RO, Department of Mathematics
University of Blida
B.P. 270, Blida, Algeria

Nader Jafari Rad

Department of Mathematics, Shahrood University of Technology
Shahrood, Iran
and
School of Mathematics
Institute for Research in Fundamental Sciences (IPM),
P.O. Box 19395-5746, Tehran, Iran

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 an RDF is the value f(V(G)) = ∑u ∈ V(G)f(u). An RDF f in a graph G is independent if no two vertices assigned positive values are adjacent. The Roman domination number γR(G) (respectively, the independent Roman domination number iR(G)) is the minimum weight of an RDF (respectively, independent RDF) on G.  We say that  γR(G) strongly equals iR(G), denoted by γR(G) ≡ iR(G), if every RDF on G of minimum weight is independent. In this paper we provide a constructive characterization of trees T with γR(T) ≡ iR(T).

Keywords: Roman domination, independent Roman domination, strong equality, trees

2010 Mathematics Subject Classification: 05C69.

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Received 8 December 2010
Revised 24 November 2011
Accepted 5 April 2012