Theoretical graph analysis demonstrates exact structure and substructure fault diameters in n-dimensional hypercubes, indicating robust network transmission delay bounds under clustered failures.
Structure connectivity and substructure connectivity are innovative indicators for assessing network reliability and fault tolerance. Similarly, fault diameter evaluates fault tolerance and transmission delays in networks. This paper extends the concept of fault diameter by introducing two new variants: structure fault diameter and substructure fault diameter, derived from structure connectivity and substructure connectivity respectively. For a connected graph G with W-structure connectivity $κ(G;W)$ or W-substructure connectivity κˢ(G;W), the W-structure fault diameter Df(G;W) and W-substructure fault diameter Dfˢ(G;W) are defined as the maximum diameter of any subgraph of G resulting from removing up to $κ(G;W)-1$ W-structures or κˢ(G;W)-1 W-substructures. For the n-dimensional hypercube Qₙ with n ≥ 3 and 1 ≤ m ≤ n - 2, we determine both Df(Qₙ;Qₘ) and Dfˢ(Qₙ;Q₁). These findings generalize existing results for the diameter and fault diameter of Qₙ, providing a broader understanding of the hypercube's structural properties under fault conditions.
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Zhao et al. (2026) studied this question.
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