TY  - JOUR
ID  - Alikhani2010
LB  - Alikhani2010
AU  - Alikhani, Saeid
AU  - Peng, Yee-hock
TI  - Dominating sets and domination polynomials of certain graphs, II
ST  - Dominating sets and domination polynomials of certain graphs, II
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2010
VL  - 30
IS  - 1
SP  - 37
EP  - 51
DO  - http://dx.doi.org/10.7494/OpMath.2010.30.1.37
UR  - http://dx.doi.org/10.7494/OpMath.2010.30.1.37
AB  - 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 \(G = C_n\).
KW  - domination polynomial
KW  - dominating set
KW  - cycle
KW  - theta graph
SN  - 1232-9274
ER  -