Opuscula Mathematica

Opuscula Math.
, no. 1
 (), 83-88
Opuscula Mathematica

Nearly perfect sets in the n-fold products of graphs

Abstract. The study of nearly perfect sets in graphs was initiated in [J. E. Dunbar, F. C. Harris, S. M. Hedetniemi, S. T. Hedetniemi, A. A. McRae, R. C. Laskar, Nearly perfect sets in graphs, Discrete Mathematics 138 (1995), 229-246]. Let \(S \subseteq V(G)\). We say that \(S\) is a nearly perfect set (or is nearly perfect) in \(G\) if every vertex in \(V(G)-S\) is adjacent to at most one vertex in \(S\). A nearly perfect set \(S\) in \(G\) is called \(1\)-maximal if for every vertex \(u \in V(G)-S\), \(S \cup \{u\}\) is not nearly perfect in $G$. We denote the minimum cardinality of a \(1\)-maximal nearly perfect set in \(G\) by \(n_p(G)\). We will call the \(1\)-maximal nearly perfect set of the cardinality \(n_p(G)\) an \(n_p(G)\)-set. In this paper, we evaluate the parameter \(n_p(G)\) for some \(n\)-fold products of graphs. To this effect, we determine \(1\)-maximal nearly perfect sets in the \(n\)-fold Cartesian product of graphs and in the \(n\)-fold strong product of graphs.
Keywords: dominating sets, product of graphs.
Mathematics Subject Classification: 05C69, 05C70.
Cite this article as:
Monika Perl, Nearly perfect sets in the n-fold products of graphs, Opuscula Math. 27, no. 1 (2007), 83-88
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.