Relationship between g-extra Connectivity and g-restricted Connectivity in Networks

The fault tolerance of a network can be measured by many parameters. Connectivity is a classic measurement parameter for evaluating the fault tolerance of a network. g-extra connectivity and g-restricted connectivity are generalizations of connectivity, which can better reflect the fault tolerance of a network. Specifically, the g-extra connectivity $\kappa_{g}(G)$ of a graph G is the minimum number of nodes whose removal will disconnect G, and each remaining component has no less than $g+1$ nodes. Furthermore, the g-restricted connectivity $\kappa^{g}(G)$ of G is the minimum number of nodes whose deletion results in a graph being disconnected and the minimum degree of each remaining component is at least g. In general, g-restricted connectivity is not equal to g-extra connectivity of a network. Therefore, many scholars often discuss g-restricted connectivity and g-extra connectivity with regard to different networks separately. In this paper, we show that g-restricted connectivity is equal to g-extra connectivity under some conditions. Then, the relationship we derived can be applied to some known networks such as the data center networks DCell and BCDC, multiprocessor network $(n,k)$-star. In addition, we construct a new network $H(G_{0},G_{1},G_{2};\mathbb{M})$ and prove that our result can be applied to it. In detail, we prove $\kappa^{g}(H(G_{0},G_{1},G_{2};\mathbb{M}))=\kappa_{g}(H(G_{0},G_{1},G_{2};\mathbb{M}))=n+g+1$ for any integers $n\geq 3$ and $ g\displaystyle \leq\lfloor\frac{n-2}{2}\rfloor$.