Redundant Trees in Bipartite Graphs
Abstract
:1. Introduction
2. Embedding Lemmas
- (i)
- for and , ;
- (ii)
- for and , .
- (ii′)
- for and , ;
- (a)
- ;
- (b)
- For and , ;
- (c)
- There exist and such that and admits a tree .
3. k-Extensible System
- (D1)
- For any k-connected (bipartite) graph G, ;
- (D2)
- For any and , there are at least k internally disjoint -paths in G;
- (D3)
- For any , construct a (bipartite) graph obtained from G by adding a vertex v and connecting v to at least vertices of G. Then, contains a subgraph such that and ;
- (D4)
- For any , if there exists a (bipartite) graph , a vertex and k internally disjoint -paths in such that and is as small as possible, then contains a subgraph such that , and, furthermore, if for any , then for any .
4. Redundant Trees
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
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Hong, Y.; Wu, Y.; Liu, Q. Redundant Trees in Bipartite Graphs. Mathematics 2025, 13, 1005. https://doi.org/10.3390/math13061005
Hong Y, Wu Y, Liu Q. Redundant Trees in Bipartite Graphs. Mathematics. 2025; 13(6):1005. https://doi.org/10.3390/math13061005
Chicago/Turabian StyleHong, Yanmei, Yihong Wu, and Qinghai Liu. 2025. "Redundant Trees in Bipartite Graphs" Mathematics 13, no. 6: 1005. https://doi.org/10.3390/math13061005
APA StyleHong, Y., Wu, Y., & Liu, Q. (2025). Redundant Trees in Bipartite Graphs. Mathematics, 13(6), 1005. https://doi.org/10.3390/math13061005