标题： 图中的独立联盟：存在性与特征化 显示英文标题

标题： Independent coalition in graphs: existence and characterization

摘要： 图 $G$ 中的一个独立联盟由两个不相交的顶点集合 $V_1$ 和 $V_2$ 组成，其中任何一个都不是独立支配集，但它们的并集 $V_1 \cup V_2$ 是一个独立支配集。 一个独立联盟划分，简称为$ic$-划分，在图$G$中是一个顶点划分$\pi= \lbrace V_1,V_2,\dots ,V_k \rbrace$，使得$\pi$中的每个集合$V_i$要么是一个单元素支配集，要么不是一个独立支配集但与另一个集合$V_j \in \pi$形成一个独立联盟。 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. 我们还确定了一些图类的独立联盟数，并研究了阶为$n$的图$G$与$IC(G)\in\{1,2,3,4,n\}$以及阶为$n$的树$T$与$IC(T)=n-1$。 显示英文摘要

摘要： 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 $\pi= \lbrace V_1,V_2,\dots ,V_k \rbrace$ such that each set $V_i$ of $\pi$ either is a singleton dominating set, or is not an independent dominating set but forms an independent coalition with another set $V_j \in \pi$. 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 $IC(T)=n-1$.