Connected coalitions in graphs

Abstract

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 $\psi$ = $\{V_1, V_2,..., V_k \}$ such that every set $V_i \in \psi$ 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 $V_j\in \psi$ 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 $\delta(G)=1$ and with no full vertex, it holds that $CC(G)

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图中的连通联盟

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支配集，但其并集$V_{1}\cup V_{2}$是连通支配集。$n=|V|$阶图$G$中的连通联盟分区是顶点分区$\psi$=$\{V_1，V_2，…，V_k\}$，使得每个集合$V_i\in\psi$要么是由一个阶为$n-1$的单顶点组成的连通支配集，要么不是连通支配集而是与另一个不是连通支配集合的集合$V_j\in\psi形成连通联盟。连接联盟数，用$CC（G）$表示，是$G$的连接联盟分区的最大基数。本文首先对图中的连通联盟进行了研究，并给出了一些基本结果。准确地说，我们刻画了所有具有连通联盟划分的图。此外，我们证明了对于任何$n$阶的图$G$，$\delta（G）=1$并且没有全顶点，它认为$CC（G）

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