The cardinality of a largest independent set of G, denoted by $\alpha \left(G\right)$, is called the independence number of G. The independent domination number $i\left(G\right)$ of a graph G is the cardinality of a smallest independent dominating set of G. We introduce the concept of the common independence number of a graph G, denoted by ${\alpha }_c\left(G\right)$, as the greatest integer r such that every vertex of G belongs to some independent subset X of $V_G$ with $\left|X\right|\ge r$. The common independence number ${\alpha }_c\left(G\right)$ of G is the limit of symmetry in G with respect to the fact that each vertex of G belongs to an independent set of cardinality ${\alpha }_c\left(G\right)$ in G, and there are vertices in G that do not belong to any larger independent set in G. For any graph G, the relations between above parameters are given by the chain of inequalities $i\left(G\right)\le {\alpha }_c\left(G\right)\le \alpha \left(G\right)$. In this paper, we characterize the trees T for which $i\left(T\right)={\alpha }_c\left(T\right)$, and the block graphs G for which ${\alpha }_c\left(G\right)=\alpha \left(G\right)$.

中文翻译：

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图中的共同独立性

G的最大独立集的基数，表示为$\alpha \left(G\right)$，称为G的独立数。独立支配号$\mathrm{一世}\left(G\right)$的曲线图的G ^是一个最小的独立控制集的基数ģ。我们引入了图G的公共独立数的概念，表示为${\alpha }_C\left(G\right)$作为最大的整数- [R ，使得每个顶点ģ属于一些独立的子集X的${伏}_G$ 和 $\left|X\right|\ge r$. 公共独立数${\alpha }_C\left(G\right)$的G ^是在对称的限制ģ相对于该各顶点的事实ģ属于一组独立的基数的${\alpha }_C\left(G\right)$在G 中，并且G 中有顶点不属于G中任何更大的独立集合。 对于任何图G，上述参数之间的关系由不等式链给出$\mathrm{一世}\left(G\right)\le {\alpha }_C\left(G\right)\le \alpha \left(G\right)$. 在本文中，我们刻画树木牛逼了这$\mathrm{一世}\left(吨\right)={\alpha }_C\left(吨\right)$，以及块图G，其中${\alpha }_C\left(G\right)=\alpha \left(G\right)$.