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arXiv 1802.02660math.CO
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3-连通图中的可删边及其应用

Deletable edges in 3-connected graphs and their applications

S. R. Kingan

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AI总结：

本文研究3-连通图中H-可删边的结构，证明无H-可删边的图可通过删除边、收缩边或删除3度顶点等操作从H逐步生成，并据此给出强分裂定理的图论证明、Dirac棱柱图刻画的新证明及Halin极小3-连通图边数上界的推广。

摘要：

设 $G$ 和 $H$ 为简单 3-连通图，且 $G$ 含有 $H$-minor。若 $G\backslash e$ 是 3-连通的且含有 $H$-minor，则称 $G$ 中的边 $e$ 为 $H$-可删边。本文的主要结果证明：若 $G$ 没有 $H$-可删边，则存在一列简单 3-连通图 $G_0, \dots, G_k$，它们都没有 $H$-可删边，使得 $G_0\cong H$，$G_k= G$，且对 $1 \le i \le k$，以下三种情形之一成立：$G_{i-1}= G_i/f$；$G_{i-1}=G_i/f \backslash e$，其中 $e$ 和 $f$ 与 $G_i$ 中的一个 3 度顶点相关联；或者 $G_{i-1}=G_i-w$，其中 $w$ 是 $G_i$ 中的一个 3 度顶点。本文给出了若干应用，包括对拟阵理论中称为强分裂定理的结果给出图论证明，对 Dirac 关于不含棱柱图 minor 的 3-连通图的刻画给出一个新的简短证明，以及对 Halin 关于极小 3-连通图边数上界的结果给出推广。Halin 证明：若 $G$ 是 $n\ge 8$ 个顶点上的极小 3-连通图，则 $|E(G)|\le 3n-9$，且等号成立当且仅当 $G\cong K_{3, n-3}$。我们给出 Halin 结果的另一种证明，并通过确定满足 $|E(G)|=3n-10$ 的极小 3-连通无限图族来推广该结果。

英文摘要：

Let $G$ and $H$ be simple 3-connected graphs such that $G$ has an $H$-minor. An edge $e$ in $G$ is called {\it $H$-deletable} if $G\backslash e$ is 3-connected and has an $H$-minor. The main result in this paper establishes that, if $G$ has no $H$-deletable edges, then there exists a sequence of simple 3-connected graphs $G_0, \dots , G_k$ with no $H$-deletable edges such that $G_0\cong H$, $G_k= G$, and for $1 \le i \le k$ one of three possibilities holds: $G_{i-1}= G_i/f$; $G_{i-1}=G_i/f \backslash e$ where $e$ and $f$ are incident to a degree 3 vertex in $G_i$; or $G_{i-1}=G_i-w$ where $w$ is a degree $3$ vertex in $G_i$. Several applications are given including a graph theoretic proof of the matroid theory result known as the Strong Splitter Theorem, a short new proof of Dirac's characterization of 3-connected graphs with no minor isomorphic to the prism graph, and an extension of a result by Halin that bounds the number of edges in a minimally 3-connected graph. Halin proved that if $G$ is a minimally $3$-connected graph on $n\ge 8$ vertices, then $|E(G)|\le 3n-9$ and equality holds if and only if $G\cong K_{3, n-3}$. We give a different proof of Halin's result and extend it by identifying the minimally 3-connected infinite family of graphs with $|E(G)|=3n-10$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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