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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2412.09885 |
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| _version_ | 1866914188241141760 |
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| author | Zhao, Honggang Sabir, Eminjan Lin, Cheng-Kuan |
| author_facet | Zhao, Honggang Sabir, Eminjan Lin, Cheng-Kuan |
| contents | 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 $n \geq 3$ and $1 \leq m \leq n - 2$, we determine both $D_f(Q_n;Q_m)$ and $D_f^s(Q_n;Q_1)$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_09885 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Structure fault diameter of hypercubes Zhao, Honggang Sabir, Eminjan Lin, Cheng-Kuan Combinatorics 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 $n \geq 3$ and $1 \leq m \leq n - 2$, we determine both $D_f(Q_n;Q_m)$ and $D_f^s(Q_n;Q_1)$. 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. |
| title | Structure fault diameter of hypercubes |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2412.09885 |