Fair coalition in graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911164718383104 |
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| author | Alikhani, Saeid Jafari, Abbas Safazadeh, Maryam |
| author_facet | Alikhani, Saeid Jafari, Abbas Safazadeh, Maryam |
| contents | 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 a smallest dominating set of $G$, denoted by $γ(G)$, 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 $k\geq 1$. We consider $1FD$-sets and define a fair coalition in a graph $G$ as a pair of disjoint subsets $A_1, A_2 \subseteq A$ that satisfy the following conditions: (a) neither $A_1$ nor $A_2$ constitutes a $1$-fair dominating set of $G$, and (b) $A_1\cup A_2$ constitutes a $1$-fair dominating set of $G$. A fair coalition partition of a graph $G$ is a partition $Υ= \{A_1,A_2,\ldots,A_k\}$ of its vertex set such that every set $A_i$ of $Υ$ is either a singleton $1$-fair dominating set of $G$, or is not a $1$-fair dominating set of $G$ but forms a fair coalition with another non-$1$-fair dominating set $A_j\in Υ$. We define the fair coalition number of $G$ as the maximum cardinality of a fair coalition partition of $G$, and we denote it by $\mathcal{C}_f(G)$. We initiate the study of the fair coalition in graphs and obtain $\mathcal{C}_f(G)$ for some specific graphs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_15080 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fair coalition in graphs Alikhani, Saeid Jafari, Abbas Safazadeh, Maryam Combinatorics 05C25, 05C60 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 a smallest dominating set of $G$, denoted by $γ(G)$, 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 $k\geq 1$. We consider $1FD$-sets and define a fair coalition in a graph $G$ as a pair of disjoint subsets $A_1, A_2 \subseteq A$ that satisfy the following conditions: (a) neither $A_1$ nor $A_2$ constitutes a $1$-fair dominating set of $G$, and (b) $A_1\cup A_2$ constitutes a $1$-fair dominating set of $G$. A fair coalition partition of a graph $G$ is a partition $Υ= \{A_1,A_2,\ldots,A_k\}$ of its vertex set such that every set $A_i$ of $Υ$ is either a singleton $1$-fair dominating set of $G$, or is not a $1$-fair dominating set of $G$ but forms a fair coalition with another non-$1$-fair dominating set $A_j\in Υ$. We define the fair coalition number of $G$ as the maximum cardinality of a fair coalition partition of $G$, and we denote it by $\mathcal{C}_f(G)$. We initiate the study of the fair coalition in graphs and obtain $\mathcal{C}_f(G)$ for some specific graphs. |
| title | Fair coalition in graphs |
| topic | Combinatorics 05C25, 05C60 |
| url | https://arxiv.org/abs/2507.15080 |