Structure fault diameter of hypercubes

2024/12/13 by Honggang Zhao, Zhao, Honggang, Eminjan Sabir +4
Computer Science · Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Interconnection Networks and Systems #math.CO

paper · pdf · doi:10.48550/arxiv.2412.09885

openalex publication_date 2024/12/13 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28 · arxiv created 2026/07/30 · arxiv updated 2026/07/31

Abstract

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 κ^s(G;W), the W-structure fault diameter D_f(G;W) and W-substructure fault diameter D_f^s(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 κ^s(G;W)-1 W-substructures. For the n-dimensional hypercube Q_n with n ≥ 3 and 1 ≤ m ≤ n - 2, we determine both D_f(Q_n;Q_m) and D_f^s(Q_n;Q₁). These findings generalize existing results for the diameter and fault diameter of Q_n, providing a broader understanding of the hypercube's structural properties under fault conditions.

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