arXiv 2302.057542023-02-11math.CO

图中的连通联盟

Connected coalitions in graphs

Saeid Alikhani, Davood Bakhshesh, Hamidreza Golmohammadi, Elena V. Konstantinova

备注：: 14 pages, 4 figures

摘要：

图$G=(V,E)$中的连通联盟由两个不相交的顶点集$V_{1}$和$V_{2}$组成，这两个顶点集都不是连通控制集，但它们的并集$V_{1}\cup V_{2}$是连通控制集。一个$n=|V|$阶图$G$中的连通联盟划分是顶点划分$ψ$ = $\{V_1, V_2,..., V_k \}$使得每个集合$V_i \in ψ$或者是由度为$n-1$的单个顶点组成的连通控制集，或者不是连通控制集但与另一个不是连通控制集的集合$V_j\in ψ$形成连通联盟。连接联盟数，用$CC(G)$表示，是$G$的连接联盟分区的最大基数。本文对图中的连通联盟进行了研究，并给出了一些基本结果。准确地说，我们描述了所有的图，有一个连接的联盟分区。此外，我们还证明了，对于任何具有$δ(G)=1$的$n$阶图$G$，它保持$CC(G)<n$。此外，我们表明，对于任何树$T$，$CC(T)=2$。最后，我们提出了两个多项式时间算法，对于一个给定的连接图$G$的顺序$n$确定是否$CC(G)=n$或$CC(G)=n-1$。

英文摘要：

The connected coalition in a graph $G=(V,E)$ consists of two disjoint sets of vertices $V_{1}$ and $V_{2}$, neither of which is a connected dominating set but whose union $V_{1}\cup V_{2}$, is a connected dominating set. A connected coalition partition in a graph $G$ of order $n=|V|$ is a vertex partition $ψ$ = $\{V_1, V_2,..., V_k \}$ such that every set $V_i \in ψ$ either is a connected dominating set consisting of a single vertex of degree $n-1$, or is not a connected dominating set but forms a connected coalition with another set $V_j\in ψ$ which is not a connected dominating set. The connected coalition number, denoted by $CC(G)$, is the maximum cardinality of a connected coalition partition of $G$. In this paper, we initiate the study of connected coalition in graphs and present some basic results. Precisely, we characterize all graphs that have a connected coalition partition. Moreover, we show that for any graph $G$ of order $n$ with $δ(G)=1$ and with no full vertex, it holds that $CC(G)<n$. Furthermore, we show that for any tree $T$, $CC(T)=2$. Finally, we present two polynomial-time algorithms that for a given connected graph $G$ of order $n$ determine whether $CC(G)=n$ or $CC(G)=n-1$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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