Minimum edge cuts of hypercubes and k-ary n-cubes leaving every fault-free node in given subnetworks

Abstract

The hypercube and the $k$-ary $n$-cube are two leading interconnection networks for multiprocessor systems. {\it A $t$-embedded edge cut} of an $n$-dimensional recursive interconnection network $G_n$ is a set of edges whose removal separates $G_n$ and each vertex lies in a $t$-dimensional subnetwork of the residual network. The network $G_n$ is {\it super $t$-embedded edge connected} ({\it super-$\eta_t$} for short) if the deletion of an arbitrary minimum $t$-embedded edge cut of $G_n$ isolates a $t$-dimensional subnetwork $G_t$. In this note, we show that both the $n$-dimensional hypercube and the ternary $n$-cube are super-$\eta_t$ for $0\leq t\leq n-1$ except for the $2$D hypercube when $t=0$, and the $k$-ary $n$-cube with odd $k\geq 5$ is not super-$\eta_{n-1}$. An algorithm is proposed to obtain a minimum embedded-edge cut for hypercubes and $k$-ary $n$-cubes with odd $k\geq 3$.