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

中文翻译：

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平衡超立方体的结构容错

设 H 是给定图 G 的连通子图。 G 的 H 结构连通性是 G 的子图的最小集合 $${\mathcal {F}}$$ 的基数，使得 $${\ mathcal {F}}$$ 与 H 同构，移除 $${\mathcal {F}}$$ 的所有元素将断开 G。图 G 的 H 子结构连通性是最小集合的基数$${\mathcal {F}}'$$ G 的子图，使得 $${\mathcal {F}}'$$ 中的每个元素与 H 的连通子图同构，并且去除$${\mathcal {F}}'$$ 将断开 G。这两个参数是 Lin 等人提出的。在 (Theor Comput Sci 634:97–107, 2016) 中，其中对 $${\mathcal {F}}$$ 和 $${\mathcal {F}}'$$ 没有限制。在 Lu and Wu (Bull Malays Math Sci Soc 43(3):2659–2672, 2020)，对于 n 维平衡超立方体 $$\text {BH}_n$$，作者对 $${\mathcal {F}}$$（分别为 $${\mathcal {F}}'$$ ）施加了一些限制并要求 $${\mathcal {F}}$$ 中的两个元素（分别为 $${\mathcal {F}}'$$ ）不能共享一个顶点。在这样的限制下，他们确定了 $$\text {BH}_n$$ 的（受限）H-结构和（受限）H-子结构连通性，用于 $$H\in \{K_1,K_{1,1},K_ {1,2},K_{1,3},C_4\}$$ 。在本文中，我们遵循 (2016) 对两个参数的定义，并确定 $$\text {BH}_n$$ for $$H\in \{K_{1, t},P_k,C_4\}$$ ，其中 $$K_{1,t}$$ 是 $$t+1$$ 顶点上的星星，其中 $$1\le t\le 2n$$ 和 $$P_k$ $ 是长度为 k 的路径，其中 $$1\le k\le 7$$ 。我们的一些主要结果表明 H 结构连通性（resp。