Structure fault tolerance of balanced hypercubes

Let H be a connected subgraph of a given graph G. The H-structure connectivity of G is the cardinality of a minimal set \({\mathcal {F}}\) of subgraphs of G such that every element in \({\mathcal {F}}\) is isomorphic to H, and the removal of all the elements of \({\mathcal {F}}\) will disconnect G. The H-substructure connectivity of graph G is the cardinality of a minimal set \({\mathcal {F}}'\) of subgraphs of G such that every element in \({\mathcal {F}}'\) is isomorphic to a connected subgraph of H, and the removal of all the elements of \({\mathcal {F}}'\) will disconnect G. The two parameters were proposed by Lin et al. in (Theor Comput Sci 634:97–107, 2016), where no restrictions on \({\mathcal {F}}\) and \({\mathcal {F}}'\). In Lü and Wu (Bull Malays Math Sci Soc 43(3):2659–2672, 2020), the authors imposed some restrictions on \({\mathcal {F}}\) (resp. \({\mathcal {F}}'\)) for the n-dimensional balanced hypercube \(\text {BH}_n\) and requires that two elements in \({\mathcal {F}}\) (resp. \({\mathcal {F}}'\)) cannot share a vertex. Under such restrictions, they determined the (restricted) H-structure and (restricted) H-substructure connectivity of \(\text {BH}_n\) for \(H\in \{K_1,K_{1,1},K_{1,2},K_{1,3},C_4\}\). In this paper, we follow (2016) for the definitions of the two parameters and determine the H-structure and H-substructure connectivity of \(\text {BH}_n\) for \(H\in \{K_{1,t},P_k,C_4\}\), where \(K_{1,t}\) is the star on \(t+1\) vertices with \(1\le t\le 2n\) and \(P_k\) is a path of length k with \(1\le k\le 7\). Some of our main results show that the H-structure connectivity (resp. H-substructure connectivity) of \(\text {BH}_n\) is equal to the restricted H-structure connectivity (resp. restricted H-substructure connectivity) of \(\text {BH}_n\) for \(H\in \{K_{1,1},K_{1,2},C_4\}\), but the \(K_{1,3}\)-structure connectivity (resp. \(K_{1,3}\)-substructure connectivity) of \(\text {BH}_n\) is not equal to the restricted \(K_{1,3}\)-structure connectivity (resp. restricted \(K_{1,3}\)-substructure connectivity) of \(\text {BH}_n\) unless \(n=\lceil 2n/3\rceil\).