Structure Fault Tolerance of Exchanged Hypercube

Abstract The undirected graph, exchanged hypercube $EH(s,t)$, is a variant of hypercube proposed by Loh et al. It is obtained by removing some links from $(s+t+1)$-dimensional hypercube. It retains many excellent properties, so many people have studied its reliability and fault tolerance. In this paper, combining the structure connectivity and substructure connectivity of graphs proposed not long ago, we obtain its $P_k$-path, $C_{2l}$-cycle and $K_{1,r}$-star structure connectivity and substructure connectivity where $2\le k,r\le s-1\le t-1$ and $6\le 2l\le s-1\le t-1$; we also establish $\kappa ^s(EH(s,t);C_4)$ for $5\le s\le t$ and the upper bound of $\kappa (EH(s,t);C_4)$ for $4\le s\le t$.