On $g$-Extra Connectivity of Hypercube-like Networks

Given a connected graph $G$ and a non-negative integer $g$, the {\em $g$-extra connectivity} $\k_g(G)$ of $G$ is the minimum cardinality of a set of vertices in $G$, if it exists, whose deletion disconnects $G$ and leaves each remaining component with more than $g$ vertices. This paper focuses on the $g$-extra connectivity of hypercube-like networks (HL-networks for short) which includes numerous well-known topologies, such as hypercubes, twisted cubes, crossed cubes and M\"obius cubes. All the known results suggest the equality $\k_g(X_n)=f_n(g)$ holds, where $X_n$ is an $n$-dimensional HL-network, $f_n(g)=n(g+1)-\frac{g(g+3)}{2}$, $n\geq 5$ and $0\leq g\leq n-3$? Some authors also attempted to prove this equality in general. In this paper, we construct a subfamily of an $n$-dimensional HL-network with $g$-extra connectivity greater than $f_n(g)$ which implies that the above equality does not hold in general. We also prove that for $n\geq 5$ and $0\leq g\leq n-3$, $\k_g(X_n)\geq f_n(g)$ always holds. This enables us to give a sufficient condition for the equality $\k_g(X_n)=f_n(g)$, which is then used to determine the $g$-extra connectivity of HL-networks for some small $g$ or the $g$-extra connectivity of some particular subfamily of HL-networks. As a result, a short proof for the main results in [Journal of Computer and System Sciences 79 (2013) 669--688].