Fault-Tolerant Path-Embedding of Twisted Hypercube-Like Networks THLNs

The twisted hypercube-like networks($THLNs$) contain several important hypercube variants. This paper is concerned with the fault-tolerant path-embedding of $n$-dimensional($n$-$D$) $THLNs$. Let $G_n$ be an $n$-$D$ $THLN$ and $F$ be a subset of $V(G_n)\cup E(G_n)$ with $|F|\leq n-2$. We show that for arbitrary two different correct vertices $u$ and $v$, there is a faultless path $P_{uv}$ of every length $l$ with $2^{n-1}-1\leq l\leq 2^n-f_v-1-\alpha$, where $\alpha=0$ if vertices $u$ and $v$ form a normal vertex-pair and $\alpha=1$ if vertices $u$ and $v$ form a weak vertex-pair in $G_n-F$($n\geq5$).