Independent coalition in graphs: existence and characterization
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arXiv
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| Natura: | Preprint |
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2023
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| _version_ | 1866917733580406784 |
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| author | Samadzadeh, Mohammad Reza Mojdeh, Doost Ali |
| author_facet | Samadzadeh, Mohammad Reza Mojdeh, Doost Ali |
| contents | An independent coalition in a graph $G$ consists of two disjoint sets of vertices $V_1$ and $V_2$ neither of which is an independent dominating set but whose union $V_1 \cup V_2$ is an independent dominating set. An independent coalition partition, abbreviated, $ic$-partition, in a graph $G$ is a vertex partition $π= \lbrace V_1,V_2,\dots ,V_k \rbrace$ such that each set $V_i$ of $π$ either is a singleton dominating set, or is not an independent dominating set but forms an independent coalition with another set $V_j \in π$.
The maximum number of classes of an $ic$-partition of $G$ is the independent coalition number of $G$, denoted by $IC(G)$. In this paper we study the concept of $ic$-partition. In particular, we discuss the possibility of the existence of $ic$-partitions in graphs and introduce a family of graphs for which no $ic$-partition exists. We also determine the independent coalition number of some classes of graphs and investigate graphs $G$ of order $n$ with $IC(G)\in\{1,2,3,4,n\}$ and the trees $T$ of order $n$ with $IC(T)=n-1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_02079 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Independent coalition in graphs: existence and characterization Samadzadeh, Mohammad Reza Mojdeh, Doost Ali Combinatorics An independent coalition in a graph $G$ consists of two disjoint sets of vertices $V_1$ and $V_2$ neither of which is an independent dominating set but whose union $V_1 \cup V_2$ is an independent dominating set. An independent coalition partition, abbreviated, $ic$-partition, in a graph $G$ is a vertex partition $π= \lbrace V_1,V_2,\dots ,V_k \rbrace$ such that each set $V_i$ of $π$ either is a singleton dominating set, or is not an independent dominating set but forms an independent coalition with another set $V_j \in π$. The maximum number of classes of an $ic$-partition of $G$ is the independent coalition number of $G$, denoted by $IC(G)$. In this paper we study the concept of $ic$-partition. In particular, we discuss the possibility of the existence of $ic$-partitions in graphs and introduce a family of graphs for which no $ic$-partition exists. We also determine the independent coalition number of some classes of graphs and investigate graphs $G$ of order $n$ with $IC(G)\in\{1,2,3,4,n\}$ and the trees $T$ of order $n$ with $IC(T)=n-1$. |
| title | Independent coalition in graphs: existence and characterization |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2306.02079 |