Structure Fault Tolerance of Fully Connected Cubic Networks
Abstract
:1. Introduction
2. Preliminaries
2.1. Graph Definitions and Notations
2.2. Fully Connected Cubic Networks
- 1.
- has vertex set and edge set for .
- 2.
- For , the graph is constructed from eight vertex-disjoint copies of by adding 28 edges. For , let represent a copy of with each vertex being prefixed by m. Specifically, the vertex set of is and for each . Then, the graph is defined as and with .
3. Extra Connectivity, Structure Connectivity and t-Extra L-Structure Connectivity
- Case 1: . We set for each . By Lemma 6, . Since and for each , there is an edge, say , in with . Since and are both connected, is connected. Similarly, one can deduce that is connected for each . Thus, we have if .
- Case 2: . By Case 1, we have that is connected. Since , we have that is connected. We set . By Lemma 6, . Since and , there exists an edge, say , in with . From the fact that and are both connected, one can infer that is also connected for . □
3.1. Extra Connectivity
- Case 1: . Let R be the largest connected component of . If is connected, then R contains vertices. Otherwise, by Lemma 4, it contains four vertices. Consequently, R contains at least three intercubic vertices, and one of these intercubic vertices has a neighbor in . Hence, the subgraph induced by and is connected and contains at least vertices.
- Case 2: . Since , there are at least three vertices, say and , in such that those vertices are not boundary vertex of . By Lemma 6, each vertex of them has a neighbor in . Thus, the subgraph induced by contains vertices. However, . Therefore, this lemma is valid when .
3.2. Structure Connectivity
3.3. t-Extra L-Structure Connectivity
4. Extra Edge Connectivity, Structure Edge Connectivity and g-Extra H-Structure Edge Connectivity
- Case 1: . Let , and let R be the set . Since and , by Lemma 6, there is an index z in R such that the edge between and and the edge between and are not in . Since is connected, can connect to via . Consequently, for any two distinct indices and in S, there exists an index such that can connect to via . Thus, is connected if .
- Case 2: . By Case 1, we have that is connected. By Lemma 5, is connected. By Lemma 6, there are seven edges between and . Since , there is an edge between and such that it is not in . Thus, is connected if . □
4.1. Extra Edge Connectivity
- Case 1: . We have that can connect to via its corresponding intercubic edge. Hence, the subgraph H induced by R and is connected and it contains vertices.
- Case 2: . We have .
- Case 2.1: . Let . Since and y can connect to via its corresponding intercubic edge, can connect to via y. Hence, the subgraph H induced by , and is connected and it contains vertices.
- Case 2.2: . Since , we have that there are at least two edges corresponding to that are in . Since , . Similarly to Case 2.1, the subgraph H induced by R and is connected and contains vertices. □
4.2. Structure Edge Connectivity
- Case 1: . Set . Obviously, S’s every element is isomorphic to and . Since is disconnected, we have that is clearly a -structure edge cut of . This means that and .
- Case 2: . Set . Obviously, F’S every element isomorphic to and . Since is disconnected, we have that is clearly a -structure edge cut of . This implies that and .
4.3. g-Extra H-Structure Edge Connectivity
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Notation | Description |
---|---|
A graph with vertex set V and edge set E | |
A vertex subset of G | |
Neighbor set of a vertex subset of G | |
Edge neighbor set of a vertex subset of G | |
A complete graph | |
Star graph | |
Hypercube | |
n-Dimensional fully connected cubic networks | |
Connectivity of the graph G | |
Edge connectivity of the graph G | |
g-Extra connectivity of the graph G | |
g-Extra edge connectivity of the graph G | |
L-structure connectivity of the graph G | |
L-structure edge connectivity of the graph G | |
g-Extra L-structure connectivity of the graph G | |
g-Extra L-structure edge connectivity of the graph G | |
Set of natural numbers |
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Sabir, E.; Lin, C.-K. Structure Fault Tolerance of Fully Connected Cubic Networks. Mathematics 2025, 13, 1532. https://doi.org/10.3390/math13091532
Sabir E, Lin C-K. Structure Fault Tolerance of Fully Connected Cubic Networks. Mathematics. 2025; 13(9):1532. https://doi.org/10.3390/math13091532
Chicago/Turabian StyleSabir, Eminjan, and Cheng-Kuan Lin. 2025. "Structure Fault Tolerance of Fully Connected Cubic Networks" Mathematics 13, no. 9: 1532. https://doi.org/10.3390/math13091532
APA StyleSabir, E., & Lin, C.-K. (2025). Structure Fault Tolerance of Fully Connected Cubic Networks. Mathematics, 13(9), 1532. https://doi.org/10.3390/math13091532