3-连通图中保持子图的可收缩边
Contractible edges in 3-connected graphs that preserve a minor
摘要：
设$G$是一个具有$3$ -连通（或足够小）简单子图$H$的$3$ -连通图.我们建立了$G$有一个森林$F$，它至少有$\left\lceil(|G|-|H|+1)/2\right\rceil$边，使得$G/e$是$3$ -连接的，每个$e\in E(F)$都有一个$H$ -minor。此外，我们可以挑选具有$|G|-|H|$边的$F$，只要$G$是无三角形的。这些结果是尖锐的。我们的结果推广了Ando et.例如，其建立了$3$ -连通图$G$具有至少$\left\lceil|G|/2\right\rceil$可收缩边。作为另一个结果，每个无三角形的$3$连通图具有可收缩边的生成树。我们的结果遵循一个更一般的定理图未成年人，分裂定理，这也是建立在这里。
英文摘要：
Let $G$ be a $3$-connected graph with a $3$-connected (or sufficiently small) simple minor $H$. We establish that $G$ has a forest $F$ with at least $\left\lceil(|G|-|H|+1)/2\right\rceil$ edges such that $G/e$ is $3$-connected with an $H$-minor for each $e\in E(F)$. Moreover, we may pick $F$ with $|G|-|H|$ edges provided $G$ is triangle-free. These results are sharp. Our result generalizes a previous one by Ando et. al., which establishes that a $3$-connected graph $G$ has at least $\left\lceil|G|/2\right\rceil$ contractible edges. As another consequence, each triangle-free $3$-connected graph has an spanning tree of contractible edges. Our results follow from a more general theorem on graph minors, a splitter theorem, which is also established here.
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