Minimum embedded-link-cut split k-ary n-cubes

Given two integers $n$ and $t$ with $0\le t<n$, a $t$-embedded-link-cut of an $n$-dimensional recursive interconnection network $N_n$ is a set of links whose removal separates $N_n$ and each node still lies in a $t$-dimensional subnetwork $N_t$ in the resulting network. A minimum embedded-link-cut split $N_n$ is the resultant network obtained by removing all the links in a minimum embedded-link-cut from $N_n$. And $N_n$ is said to be super $t$-embedded link connected (super-${\eta }_t$) if each minimum embedded-link-cut split $N_n$ has exactly two components, one of which is isomorphic to $N_t$. The $k$-ary $n$-cube $Q_n^k$ is a leading interconnection network for the multiprocessor system, which takes the hypercube (i.e., the binary $n$-cube) as a special case. Let $\nu$ be the number of nodes in $Q_n^k$. In this paper, we provide an $O\left(\nu \cdot (n-t)\right)$ algorithm to obtain a minimum $t$-embedded-link-cut for $k$-ary $n$-cubes with $k=2$ or odd $k\ge 3$, and prove that both the binary $n$-cube with $n\ge 3$ and the ternary $n$-cube with $n\ge 1$ are super-${\eta }_t$ for $0\le t\le n-1$, but the $k$-ary $n$-cube with odd $k\ge 5$ is not super-${\eta }_{n-1}$.