极小3-连通平面图的构造
The Structure of Minimally 3-Connected Planer Graphs
DOI: 10.12677/pm.2025.155176, PDF,   
作者: 祝誉升：南宁师范大学数学与统计学院，广西 南宁
关键词: 极小3-连通平面图；结构；Minimally 3-Connected Planar Graph； Structure
摘要: 设 G 是由满足以下条件的3-连通平面二部图所组成的图类： G 的一部是3度点的集合，另外一部是度至少为4的点的集合。本文证明了若G是极小3-连通平面图且G中不存在边e使得G/e或G/e\f是极小3-连通平面图，则 G∈G ，这里f与e相邻于一个3度点。
Abstract: Let G be a set of minimally 3-connected planer graphs such that every member of G is a bipartite graph with one parts of vertices of degree three and the other parts of degree at least four. Let G be a minimally 3-connected planar graph. This paper show that if G has no edge e such that either G/e or G/e\f is minimally 3-connected planar graph then G∈G ; here e and f are two edges incident to a vertex of degree 3.
文章引用：祝誉升. 极小3-连通平面图的构造[J]. 理论数学, 2025, 15(5): 272-279. https://doi.org/10.12677/pm.2025.155176

参考文献

[1] Mader, W. (1972) Ecken Vom Gradn in Minimalenn-Fach Zusammenhangenden Graphen. Archiv der Mathematik, 23, 219-224. [Google Scholar] [CrossRef] 
[2] Ota, K. and Saito, A. (1988) Non-Separating Induced Cycles in 3-Connected Graphs. Scientia Series A, 2, 101-105.
[3] Dean, N., Hemminger, R.L. and Ota, K. (1989) Longest Cycles in 3‐Connected Graphs Contain Three Contractible Edges. Journal of Graph Theory, 13, 17-21. [Google Scholar] [CrossRef] 
[4] Dawes, R.W. (1986) Minimally 3-Connected Graphs. Journal of Combinatorial Theory, Series B, 40, 159-168. [Google Scholar] [CrossRef] 
[5] Coullard, C.R. and Oxley, J.G. (1992) Extensions of Tutte’s Wheels-And-Whirls Theorem. Journal of Combinatorial Theory, Series B, 56, 130-140. [Google Scholar] [CrossRef] 
[6] Kingan, S.R. (2023) Deletable Edges in 3-Connected Graphs and Their Applications. arXiv:1 802.02660.
[7] Dirac, G.A. (1963) Some Results Concerning the Structure of Graphs. Canadian Mathematical Bulletin, 6, 183-210. [Google Scholar] [CrossRef] 
[8] Mader, W. (1988) Generalizations of Critical Connectivity of Graphs. Annals of Discrete Mathematics, 38, 267-283. [Google Scholar] [CrossRef] 
[9] Halin, R. (1969) Zur Theorie Dern-Fach Zusammenhängenden Graphen. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 33, 133-164. [Google Scholar] [CrossRef] 
[10] Qin, C., Geng, J., Yang, H. and Xie, X. (2025) Contractible Edges in Spanning Trees of 3-Connected Graphs. Graphs and Combinatorics, 41, Article No. 22. [Google Scholar] [CrossRef] 

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