Structure and substructure connectivity of folded divide-and-swap cube

Abstract

Let $ H $ be a connected subgraph of a graph $ G $. The structure connectivity of $ G $, denoted by $ \kappa(G;H) $, is the minimum number of a set of connected subgraphs in $ G $, whose removal disconnects $ G $ and each element in the set is isomorphic to $ H $. The substructure connectivity of $ G $, denoted by $ \kappa^s(G;H) $, is the minimum number of a set of connected subgraphs in $ G $, whose removal disconnects $ G $ and each element in the set is isomorphic to a connected subgraph of $ H $. In this paper, we determine $ H $-structure connectivity and $ H $-substructure connectivity of folded divide-and-swap cube $ FDSC_n $ for $ H\in\{K_1, K_{1,1}, K_{1,m} (2\leq m \leq d+1) \} $ where $ n=2^d $. We show that $\kappa(FDSC_n;K_1)=\kappa^s(FDSC_n;K_1)=d+2$, $\kappa(FDSC_n;K_{1,1})=\kappa^s(FDSC_n;K_{1,1})=d+1 $ for $ d\geq1 $ and $\kappa(FDSC_n;K_{1,m})=\kappa^s(FDSC_n;K_{1,m})=\lfloor\frac{d}{2}\rfloor+1$ for $d\geq1 $ and $ 2\leq m \leq d+1$.