On the $g$-good-neighbor connectivity of graphs

Abstract

Connectivity and diagnosability are two important parameters for the fault tolerant of an interconnection network $G$. In 1996, Fà brega and Fiol proposed the $g$-good-neighbor connectivity of $G$. In this paper, we show that $1\le \hat{I}{º}^g(G)\le n-2g-2$ for $0\le g\le \left\{\hat{I}”(G),\lfloor \frac{n-3}{2}\rfloor \right\}$, and graphs with $\hat{I}{º}^g(G)=1,2$ and trees with $\hat{I}{º}^g(T_n)=n-t$ for $4\le t\le \frac{n+2}{2}$ are characterized, respectively. In the end, we get the three extremal results for the $g$-good-neighbor connectivity.

14 pages, 2 figures. arXiv admin note: substantial text overlap with arXiv:1904.06527; text overlap with arXiv:1609.08885, arXiv:1612.05381 by other authors