Let $G$ be graph with vertex set $V$ and order $n=|V|$. A coalition in $G$ is a combination of two distinct sets, $A\subseteq V$ and $B\subseteq V$, which are disjoint and are not dominating sets of $G$, but $A\cup B$ is a dominating set of $G$. A coalition partition of $G$ is a partition $\mathcal{P}=\{S_1,\ldots,S_k\}$ of its vertex set $V$, where each set $S_i\in \mathcal{P}$ is either a dominating set of $G$ with only one vertex, or it is not a dominating set but forms a coalition with some other set $S_j \in \mathcal{P}$. The coalition number $C(G)$ is the maximum cardinality of a coalition partition of $G$. To represent a coalition partition $\mathcal{P}$ of $G$, a coalition graph $\CG(G, \mathcal{P})$ is created, where each vertex of the graph corresponds to a member of $\mathcal{P}$ and two vertices are adjacent if and only if their corresponding sets form a coalition in $G$. A coalition partition $\mathcal{P}$ of $G$ is a singleton coalition partition if every set in $\mathcal{P}$ consists of a single vertex. If a graph $G$ has a singleton coalition partition, then $G$ is referred to as a singleton-partition graph. A graph $H$ is called a singleton coalition graph of a graph $G$ if there exists a singleton coalition partition $\mathcal{P}$ of $G$ such that the coalition graph $\CG(G,\mathcal{P})$ is isomorphic to $H$. A singleton coalition graph chain with an initial graph $G_1$ is defined as the sequence $G_1\rightarrow G_2\rightarrow G_3\rightarrow\cdots$ where all graphs $G_i$ are singleton-partition graphs, and $\CG(G_i,\Gamma_1)=G_{i+1}$, where $\Gamma_1$ represents a singleton coalition partition of $G_i$. In this paper, we address two open problems posed by Haynes et al. We characterize all graphs $G$ of order $n$ and minimum degree $\delta(G)=2$ such that $C(G)=n$ and investigate the singleton coalition graph chain starting with graphs $G$ where $\delta(G)\le 2$.

翻译：设$G$是一个图，顶点集为$V$，阶为$n=|V|$。$G$中的联盟是指两个互不相交且均非$G$的支配集的不同子集$A\subseteq V$和$B\subseteq V$，但$A\cup B$是$G$的支配集。$G$的联盟划分是其顶点集$V$的一个划分$\mathcal{P}=\{S_1,\ldots,S_k\}$，其中每个集合$S_i\in \mathcal{P}$要么是仅含一个顶点的$G$的支配集，要么不是支配集但与某个其他集合$S_j \in \mathcal{P}$形成联盟。联盟数$C(G)$是$G$的联盟划分的最大基数。为表示$G$的联盟划分$\mathcal{P}$，构造联盟图$\CG(G, \mathcal{P})$，该图的每个顶点对应$\mathcal{P}$的一个成员，且两个顶点相邻当且仅当它们在$G$中对应的集合形成联盟。若$\mathcal{P}$中的每个集合均由单个顶点组成，则称$\mathcal{P}$为$G$的单顶点联盟划分。若图$G$存在单顶点联盟划分，则称$G$为单顶点划分图。若存在$G$的单顶点联盟划分$\mathcal{P}$使得联盟图$\CG(G,\mathcal{P})$同构于$H$，则称图$H$为$G$的单顶点联盟图。以初始图$G_1$为起点的单顶点联盟图链定义为序列$G_1\rightarrow G_2\rightarrow G_3\rightarrow\cdots$，其中所有图$G_i$均为单顶点划分图，且$\CG(G_i,\Gamma_1)=G_{i+1}$，这里$\Gamma_1$表示$G_i$的一个单顶点联盟划分。本文解决了Haynes等人提出的两个开放问题。我们刻画了所有阶为$n$且最小度$\delta(G)=2$满足$C(G)=n$的图$G$，并研究了以$\delta(G)\le 2$的图$G$为起点的单顶点联盟图链。