Opuscula Math. 31, no. 2 (2011), 153-158
http://dx.doi.org/10.7494/OpMath.2011.31.2.153

 
Opuscula Mathematica

A note on global alliances in trees

Mohamed Bouzefrane
Mustapha Chellali

Abstract. For a graph \(G=(V,E)\), a set \(S\subseteq V\) is a dominating set if every vertex in \(V-S\) has at least a neighbor in \(S\). A dominating set \(S\) is a global offensive (respectively, defensive) alliance if for each vertex in \(V-S\) (respectively, in \(S\)) at least half the vertices from the closed neighborhood of \(v\) are in \(S\). The domination number \(\gamma(G)\) is the minimum cardinality of a dominating set of \(G\), and the global offensive alliance number \(\gamma_{o}(G)\) (respectively, global defensive alliance number \(\gamma_{a}(G)\)) is the minimum cardinality of a global offensive alliance (respectively, global deffensive alliance) of \(G\). We show that if \(T\) is a tree of order \(n\), then \(\gamma_{o}(T)\leq 2\gamma(T)-1\) and if \(n\geq 3\), then \(\gamma_{o}(T)\leq \frac{3}{2}\gamma_{a}(T)-1\). Moreover, all extremal trees attaining the first bound are characterized.

Keywords: global defensive alliance, global offensive alliance, domination, trees.

Mathematics Subject Classification: 05C69.

Full text (pdf)

  • Mohamed Bouzefrane
  • University of Blida, LAMDA-RO Laboratory, Department of Mathematics, B.P. 270, Blida, Algeria
  • Mustapha Chellali
  • University of Blida, LAMDA-RO Laboratory, Department of Mathematics, B.P. 270, Blida, Algeria
  • Received: 2009-11-11.
  • Revised: 2010-08-15.
  • Accepted: 2010-08-18.
Opuscula Mathematica - cover

Cite this article as:
Mohamed Bouzefrane, Mustapha Chellali, A note on global alliances in trees, Opuscula Math. 31, no. 2 (2011), 153-158, http://dx.doi.org/10.7494/OpMath.2011.31.2.153

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.