Opuscula Math. 33, no. 4 (2013), 641-646

Opuscula Mathematica

A note on k-Roman graphs

Ahmed Bouchou
Mostafa Blidia
Mustapha Chellali

Abstract. Let \(G=\left(V,E\right)\) be a graph and let \(k\) be a positive integer. A subset \(D\) of \(V\left( G\right) \) is a \(k\)-dominating set of \(G\) if every vertex in \(V\left( G\right) \backslash D\) has at least \(k\) neighbours in \(D\). The \(k\)-domination number \(\gamma_{k}(G)\) is the minimum cardinality of a \(k\)-dominating set of \(G.\) A Roman \(k\)-dominating function on \(G\) is a function \(f\colon V(G)\longrightarrow\{0,1,2\}\) such that every vertex \(u\) for which \(f(u)=0\) is adjacent to at least \(k\) vertices \(v_{1},v_{2},\ldots ,v_{k}\) with \(f(v_{i})=2\) for \(i=1,2,\ldots ,k.\) The weight of a Roman \(k\)-dominating function is the value \(f(V(G))=\sum_{u\in V(G)}f(u)\) and the minimum weight of a Roman \(k\)-dominating function on \(G\) is called the Roman \(k\)-domination number \(\gamma_{kR}\left( G\right)\) of \(G\). A graph \(G\) is said to be a \(k\)-Roman graph if \(\gamma_{kR}(G)=2\gamma_{k}(G).\) In this note we study \(k\)-Roman graphs.

Keywords: Roman \(k\)-domination, \(k\)-Roman graph.

Mathematics Subject Classification: 05C69.

Full text (pdf)

  • Ahmed Bouchou
  • University Dr Yahia Fares, Médéa, Algeria
  • Mostafa Blidia
  • University of Blida, LAMDA-RO, Department of Mathematics, B.P. 270, Blida, Algeria
  • Mustapha Chellali
  • LAMDA-RO, Department of Mathematics, B.P. 270, Blida, Algeria
  • Received: 2012-10-11.
  • Revised: 2013-02-25.
  • Accepted: 2013-04-04.
Opuscula Mathematica - cover

Cite this article as:
Ahmed Bouchou, Mostafa Blidia, Mustapha Chellali, A note on k-Roman graphs, Opuscula Math. 33, no. 4 (2013), 641-646, http://dx.doi.org/10.7494/OpMath.2013.33.4.641

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.