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Let H be a connected subgraph of a connected graph G. The H-structure connectivity of the graph G, denoted by κ(G;H), is the minimum cardinality of a minimal set of subgraphs F={H1′,H2′,…,Hm′} in G, such that every H′i∈F is isomorphic to H and removal of F from G will disconnect G. The H-substructure connectivity of the graph G, denoted by κs(G;H), is the minimum cardinality of a minimal set of subgraphs F={J1′,J2′,…,Jm′} in G, such that every Ji′∈F is a connected subgraph of H and removal of F from G will disconnect G. In this paper, we provide the H-structure and the H-substructure connectivity of the circulant graph Cir(n,Ω) where Ω={1,…,k,n−k,…,n−1},1≤k≤⌊n2⌋ and the hypercube Qn for some connected subgraphs H.