Structure and substructure connectivity of balanced hypercubes

The connectivity of a network directly signifies its reliability and fault-tolerance. Structure and substructure connectivity are two novel generalizations of the connectivity. Let $H$ be a subgraph of a connected graph $G$. The structure connectivity (resp. substructure connectivity) of $G$, denoted by $\kappa(G;H)$ (resp. $\kappa^s(G;H)$), is defined to be the minimum cardinality of a set $F$ of connected subgraphs in $G$, if exists, whose removal disconnects $G$ and each element of $F$ is isomorphic to $H$ (resp. a subgraph of $H$). In this paper, we shall establish both $\kappa(BH_n;H)$ and $\kappa^s(BH_n;H)$ of the balanced hypercube $BH_n$ for $H\in\{K_1,K_{1,1},K_{1,2},K_{1,3},C_4\}$.