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Alikhani, Saeid, Bakhshesh, Davood, Jafari, Nasrin, Safazadeh, Maryam. (1401). Some new results on the number of fair dominating sets. , 54(2), 1-12. doi: 10.22059/jac.2022.90381
Saeid Alikhani; Davood Bakhshesh; Nasrin Jafari; Maryam Safazadeh. "Some new results on the number of fair dominating sets". , 54, 2, 1401, 1-12. doi: 10.22059/jac.2022.90381
Alikhani, Saeid, Bakhshesh, Davood, Jafari, Nasrin, Safazadeh, Maryam. (1401). 'Some new results on the number of fair dominating sets', , 54(2), pp. 1-12. doi: 10.22059/jac.2022.90381
Alikhani, Saeid, Bakhshesh, Davood, Jafari, Nasrin, Safazadeh, Maryam. Some new results on the number of fair dominating sets. , 1401; 54(2): 1-12. doi: 10.22059/jac.2022.90381

Some new results on the number of fair dominating sets

Journal of Algorithms and Computation
دوره 54، شماره 2، اسفند 2022، صفحه 1-12    اصل مقاله (357.87 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22059/jac.2022.90381
نویسندگان
Saeid Alikhani* 1؛ Davood Bakhshesh2؛ Nasrin Jafari3؛ Maryam Safazadeh3
1Department of Mathematical Sciences, Yazd University, 89195-741, Yazd, Iran
2Department of Computer Science, University of Bojnord, Bojnord, Iran.
3Department of Mathematical Sciences, Yazd University, 89195-741, Yazd, Iran.
چکیده
Let $G=(V, E)$ be a simple graph. A dominating set of $G$ is a subset $D\subseteq V$ such that every vertex not in $D$ is adjacent to at least one vertex in $D$.
The cardinality of the smallest dominating set of $G$, denoted by $\gamma(G)$, is the domination number of $G$. For $k \geq 1$, a $k$-fair dominating set ($kFD$-set) in $G$, is a dominating set $S$ such that $|N(v) \cap D|=k$ for every vertex $ v \in V\setminus D$. A fair dominating set in $G$ is a $kFD$-set for some integer $k\geq 1$. Let ${\cal D}_f(G,i)$ be the family of the
fair dominating sets of a graph $G$ with cardinality $i$ and let
$d_f(G,i)=|{\cal D}_f(G,i)|$.
The fair domination polynomial of $G$ is $D_f(G,x)=\sum_{ i=1}^{|V(G)|} d_f(G,i) x^{i}$. In this paper, 
after computation of the fair domination number of power of cycle, we count the number of the fair dominating sets of certain graphs such as cubic graphs of order~$10$, power of paths, and power of cycles. As a consequence, all cubic graphs of order $10$ and especially the Petersen graph are determined uniquely by their fair domination polynomial. 
کلیدواژه‌ها
fair dominating set؛ Petersen graph؛ power of path and cycle؛ unimodal
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