On the gg-good-neighbor connectivity of graphs

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

Hybrid fault diagnosis capability analysis of highly connected graphs

Zhu et al. [Theoret. Comput. Sci. 758 (2019) 1--8] introduced the $h$-edge tolerable diagnosability to measure the fault diagnosis capability of a multiprocessor system with faulty links. This kind of diagnosability is a generalization of the concept of traditional diagnosability. A graph is called a maximally connected graph if its minimum degree equals its vertex connectivity. It is well-known that many irregular networks are maximally connected graphs and the $h$-edge tolerable diagnosabilities of these networks are unknown, which is our motivation for research. In this paper, we obtain the lower bound of the $h$-edge tolerable diagnosability of a $t$-connected graph and establish the $h$-edge tolerable diagnosability of a maximally connected graph under the PMC model and the MM$^*$ model, which extends some results in [IEEE Trans. Comput. 23 (1974) 86--88], [IEEE Trans. Comput. 53 (2004) 1582--1590] and [Theoret. Comput. Sci. 796 (2019) 147--153].Comment: 20 pages, 6 figures. arXiv admin note: text overlap with arXiv:1904.0702