For any graph G, a set of vertices $\mathcal{V}$ is said to be dominating if every vertex of G contains at least one node of G and separating if each vertex v contains a unique neighbour $u_v\in \mathcal{V}$ that is adjacent to no other vertex of G. If $\mathcal{V}$ is both dominating and separating, then $\mathcal{V}$ is defined to be an identification code. In this paper, we study strong identification codes with an index r, by imposing the constraint that each vertex of G contains at least r unique neighbours in $\mathcal{V}$. We use the probabilistic method to study both the minimum size of strong identification codes and the existence of graphs that allow an identification code with a given index.

中文翻译：

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图形的强识别代码

对于任何图G，一组顶点$\mathcal{伏}$被说成是主导的，如果每个顶点ģ包含至少一个节点ģ和分离如果每个顶点v包含一个唯一的邻居${你}_v\in \mathcal{伏}$不与G 的任何其他顶点相邻。如果$\mathcal{伏}$ 既是支配又是分离，那么 $\mathcal{伏}$定义为识别码。在本文中，我们研究与指数强劲识别码[R ，通过施加的每个顶点的约束摹至少包含[R独特的邻居$\mathcal{伏}$. 我们使用概率方法来研究强识别码的最小尺寸以及允许识别码具有给定索引的图的存在性。