Opuscula Mathematica
Opuscula Math. 30, no. 1 (), 37-51
http://dx.doi.org/10.7494/OpMath.2010.30.1.37
Opuscula Mathematica

Dominating sets and domination polynomials of certain graphs, II



Abstract. The domination polynomial of a graph \(G\) of order \(n\) is the polynomial \(D(G,x) = \sum _{i=\gamma(G)}^n d(G,i)x^i\), where \(d(G,i)\) is the number of dominating sets of \(G\) of size \(i\), and \(\gamma (G)\) is the domination number of \(G\). In this paper, we obtain some properties of the coefficients of \(D(G,x)\). Also, by study of the dominating sets and the domination polynomials of specific graphs denoted by \(G^{\prime}(m)\), we obtain a relationship between the domination polynomial of graphs containing an induced path of length at least three, and the domination polynomial of related graphs obtained by replacing the path by shorter path. As examples of graphs \(G^{\prime}(m)\), we study the dominating sets and domination polynomials of cycles and generalized theta graphs. Finally, we show that, if \(n \equiv 0,2(mod\, 3)\) and \(D(G,x) = D(C_n, x)\), then \(G = C_n\).
Keywords: domination polynomial, dominating set, cycle, theta graph.
Mathematics Subject Classification: 05C69, 11B83.
Cite this article as:
Saeid Alikhani, Yee-hock Peng, Dominating sets and domination polynomials of certain graphs, II, Opuscula Math. 30, no. 1 (2010), 37-51, http://dx.doi.org/10.7494/OpMath.2010.30.1.37
 
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 http://dx.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.