An upper bound for neighbor-connectivity of graphs

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.