Structure fault tolerance of balanced hypercubes

Abstract

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 Lu 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 $$n=\lceil 2n/3\rceil$$ .