TY  - JOUR
ID  - Chellali2012
LB  - Chellali2012
AU  - Chellali, Mustapha
AU  - Rad, Nader Jafari
TI  - A note on the independent roman domination in unicyclic graphs
ST  - A note on the independent roman domination in unicyclic graphs
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2012
VL  - 32
IS  - 4
SP  - 715
EP  - 718
DO  - http://dx.doi.org/10.7494/OpMath.2012.32.4.715
UR  - http://dx.doi.org/10.7494/OpMath.2012.32.4.715
AB  - A Roman dominating function (RDF) on a graph \(G = (V;E)\) is a function \(f : V \to \{0, 1, 2\}\) 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 \(f(V(G)) = \sum _{u \in V (G)} f(u)\). An RDF \(f\) in a graph \(G\) is independent if no two vertices assigned positive values are adjacent. The Roman domination number \(\gamma _R (G)\) (respectively, the independent Roman domination number \(i_{R}(G)\)) is the minimum weight of an RDF (respectively, independent RDF) on \(G\). We say that \(\gamma _R (G)\) strongly equals \(i_R (G)\), denoted by \(\gamma _R (G) \equiv i_R (G)\), if every RDF on \(G\) of minimum weight is independent. In this note we characterize all unicyclic graphs \(G\) with \(\gamma _R (G) \equiv i_R (G)\).
KW  - Roman domination
KW  - independent Roman domination
KW  - strong equality
SN  - 1232-9274
ER  -