Dominating sets and domination polynomials of certain graphs, II

Content Provider	Directory of Open Access Journals (DOAJ)
Author	Saeid Alikhani Yee-hock Peng
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 \(G = C_n\).
Related Links	http://www.opuscula.agh.edu.pl/vol30/1/art/opuscula_math_3002.pdf
ISSN	12329274
DOI	10.7494/OpMath.2010.30.1.37
Journal	Opuscula Mathematica
Issue Number	1
Volume Number	30
Language	English
Publisher	AGH Univeristy of Science and Technology Press
Publisher Date	2010-01-01
Publisher Place	Poland
Access Restriction	Open
Subject Keyword	Applied mathematics. Quantitative methods Domination Polynomial Dominating Set Cycle Theta Graph
Content Type	Text
Resource Type	Article
Subject	Mathematics