The neighbor-connectivity of a graph G, denoted by \(\kappa _{NB}(G)\), is the least number of vertices such that removing their closed neighborhoods from G results in a graph that is empty, complete, or disconnected. In the paper, we show that for any graph G of order n, \(\kappa _{NB}(G)\le \lceil \sqrt{2n}\ \rceil -2\). We pose a conjecture that \(\kappa _{NB}(G)\le \lceil \sqrt{n}\ \rceil -1\) for a graph G of order n. For supporting it, we show that the conjecture holds for any triangle-free graphs, Cartesian, direct, lexicographic product of any two graphs.

中文翻译：

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图形的邻居连通性的上限

图 G 的邻域连通性，用 \（\kappa _{NB}（G）\） 表示，是顶点的最小数量，因此从 G 中删除它们的闭合邻域会导致图为空、完整或断开连接。在本文中，我们表明，对于任何 n 阶的图 G，\（\kappa _{NB}（G）\le \lceil \sqrt{2n}\ \rceil -2\）。我们假设 \（\kappa _{NB}（G）\le \lceil \sqrt{n}\ \rceil -1\） 对于一个 n 阶的图 G。为了支持它，我们证明该猜想适用于任何无三角形图、笛卡尔、直接、任意两个图的字典积。