Opuscula Mathematica
Opuscula Math. 29, no. 2 (), 157-164
http://dx.doi.org/10.7494/OpMath.2009.29.2.157
Opuscula Mathematica

A note on the p-domination number of trees




Abstract. Let \(p\) be a positive integer and \(G =(V(G),E(G))\) a graph. A \(p\)-dominating set of \(G\) is a subset \(S\) of \(V(G)\) such that every vertex not in \(S\) is dominated by at least \(p\) vertices in \(S\). The \(p\)-domination number \(\gamma_p(G)\) is the minimum cardinality among the \(p\)-dominating sets of \(G\). Let \(T\) be a tree with order \(n \geq 2\) and \(p \geq 2\) a positive integer. A vertex of \(V(T)\) is a \(p\)-leaf if it has degree at most \(p-1\), while a \(p\)-support vertex is a vertex of degree at least \(p\) adjacent to a \(p\)-leaf. In this note, we show that \(\gamma_p(T) \geq (n + |L_p(T)|-|S_p(T)|)/2\), where \(L_p(T)\) and \(S_p(T)\) are the sets of \(p\)-leaves and \(p\)-support vertices of \(T\), respectively. Moreover, we characterize all trees attaining this lower bound.
Keywords: \(p\)-domination number, trees.
Mathematics Subject Classification: 05C69.
Cite this article as:
You Lu, Xinmin Hou, Jun-Ming Xu, A note on the p-domination number of trees, Opuscula Math. 29, no. 2 (2009), 157-164, http://dx.doi.org/10.7494/OpMath.2009.29.2.157
 
Download this article's citation as:
a .bib file (BibTeX), a .ris file (RefMan), a .enw file (EndNote)
or export to RefWorks.

RSS Feed

horizontal rule

ISSN 1232−9274, e-ISSN 2300−6919, DOI https://doi.org/10.7494/OpMath
Copyright © 2003−2017 OPUSCULA MATHEMATICA
Contact: opuscula@agh.edu.pl
Made by Tomasz Zabawa

horizontal rule

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.