
Structure and substructure connectivity of balanced hypercubes
The connectivity of a network directly signifies its reliability and fau...
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On the complexity of structure and substructure connectivity of graphs
The connectivity of a graph is an important parameter to measure its rel...
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Vertex Sparsification for Edge Connectivity in Polynomial Time
An important open question in the area of vertex sparsification is wheth...
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The Component Connectivity of Alternating Group Graphs and SplitStars
For an integer ℓ≥ 2, the ℓcomponent connectivity of a graph G, denoted ...
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Enterprise to Computer: Star Trek chatbot
Human interactions and humancomputer interactions are strongly influenc...
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Control of connectivity and rigidity in prismatic assemblies
How can we manipulate the topological connectivity of a threedimensiona...
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Strong Subgraph Connectivity of Digraphs: A Survey
In this survey we overview known results on the strong subgraph kconnec...
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The starstructure connectivity and starsubstructure connectivity of hypercubes and folded hypercubes
As a generalization of vertex connectivity, for connected graphs G and T, the Tstructure connectivity κ(G, T) (resp. Tsubstructure connectivity κ^s(G, T)) of G is the minimum cardinality of a set of subgraphs F of G that each is isomorphic to T (resp. to a connected subgraph of T) so that GF is disconnected. For ndimensional hypercube Q_n, Lin et al. [6] showed κ(Q_n,K_1,1)=κ^s(Q_n,K_1,1)=n1 and κ(Q_n,K_1,r)=κ^s(Q_n,K_1,r)=⌈n/2⌉ for 2≤ r≤ 3 and n≥ 3. Sabir et al. [11] obtained that κ(Q_n,K_1,4)=κ^s(Q_n,K_1,4)=⌈n/2⌉ for n≥ 6, and for ndimensional folded hypercube FQ_n, κ(FQ_n,K_1,1)=κ^s(FQ_n,K_1,1)=n, κ(FQ_n,K_1,r)=κ^s(FQ_n,K_1,r)=⌈n+1/2⌉ with 2≤ r≤ 3 and n≥ 7. They proposed an open problem of determining K_1,rstructure connectivity of Q_n and FQ_n for general r. In this paper, we obtain that for each integer r≥ 2, κ(Q_n;K_1,r)=κ^s(Q_n;K_1,r)=⌈n/2⌉ and κ(FQ_n;K_1,r)=κ^s(FQ_n;K_1,r)= ⌈n+1/2⌉ for all integers n larger than r in quare scale. For 4≤ r≤ 6, we separately confirm the above result holds for Q_n in the remaining cases.
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