The concept of neighbor connectivity originated from the assessment of the subversion of espionage networks caused by underground resistance movements, and it has now been applied to measure the disruption of networks caused by cascading failures through neighbors. In this paper, we give two necessary and sufficient conditions of the existance of $g$-good-neighbor diagnosability. We introduce a new concept called $g$-good neighbor cut-component number (gc number for short), which has close relation with $g$-good-neighbor diagnosability. Sharp lower and upper bounds of the gc number of general graphs in terms of the $g$-good neighbor connectivity is given, which provides a formula to compute the $g$-good-neighbor diagnosability for general graphs (therefore for Cartesian product graphs). As their applications, we get the exact values or bounds for the gc numbers and $g$-good-neighbor diagnosability of grid, torus networks and generalized cubes.

翻译：邻连通性概念起源于对地下抵抗运动破坏间谍网络程度的评估，现已被用于衡量因邻居节点引发的级联故障对网络造成的破坏程度。本文给出了$g$-好邻可诊断性存在的两个充要条件。我们引入了一个称为$g$-好邻割分量数（简称gc数）的新概念，该概念与$g$-好邻可诊断性密切相关。我们给出了基于$g$-好邻连通性的一般图gc数的紧致上下界，这为计算一般图（进而为笛卡尔积图）的$g$-好邻可诊断性提供了公式。作为应用，我们得到了网格、环面网络及广义超立方体的gc数与$g$-好邻可诊断性的精确值或界。