On the g-Extra Connectivity of the Enhanced Hypercubes

Abstract

Abstract Reliability evaluation of interconnection networks is of significant importance to the design and maintenance of interconnection networks. The extra connectivity is an important parameter for the reliability evaluation of interconnection networks and is a generalization of the traditional connectivity. Let $g\geq 0$ be an integer and $G$ be a connected graph; the $g$-extra connectivity of $G$ is the minimum cardinality of a set of vertices in $G$, if it exists, whose removal disconnects $G$ and leaves every component with more than $g$ vertices. Determining the $g$-extra connectivity is still an unsolved problem in many interconnection networks. Let $n$, $k$ be positive integers. Let $Q_{n,k}\, (1 \leq k \leq n-1)$ denote the $(n, k)$-enhanced hypercube. In this paper, we determine the $g$-extra connectivity of $Q_{n,k}$ is $(n+1)(g+1)-\frac{g(g+3)}{2}$ for $0\leq g\leq n-k-1$, $4\leq k\leq n-5\,(n\geq 9)$. Some previous results in [Zhang, M. and Zhou, J. (2015) On g-extra connectivity of folded hypercubes. Theor. Comput. Sci., 593, 146–153.] and [Sabir, E., Mamut, A. and Vumar, E. (2019) The extra connectivity of the enhanced hypercubes. Theor. Comput. Sci., 799, 22–31.] are extended.