Opuscula Math. 32, no. 4 (2012), 715-718

Opuscula Mathematica

A note on the independent roman domination in unicyclic graphs

Mustapha Chellali
Nader Jafari Rad

Abstract. 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 \(\gamma _R (G) \equiv i_R (G)\).

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

Mathematics Subject Classification: 05C69.

Full text (pdf)

Opuscula Mathematica - cover

Cite this article as:
Mustapha Chellali, Nader Jafari Rad, A note on the independent roman domination in unicyclic graphs, Opuscula Math. 32, no. 4 (2012), 715-718, http://dx.doi.org/10.7494/OpMath.2012.32.4.715

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

In accordance with EU legislation we advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.