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Rooted topological minors on four vertices
Journal of Combinatorial Theory Series B ( IF 1.4 ) Pub Date : 2021-06-04 , DOI: 10.1016/j.jctb.2021.05.002
Koyo Hayashi , Ken-ichi Kawarabayashi

For a graph G and a set Z of four distinct vertices of G, a diamond on Z is a subgraph of G such that, for some labeling Z={v1,v2,v3,v4}, there are three internally disjoint paths P1,P2,P3 with end vertices v1,v2 with v3,v4 on P1,P2, respectively. Therefore, this yields a K4-subdivision with branch vertices on Z.

We characterize graphs G that contain no diamond on a prescribed set Z of four vertices, under the assumption that for every vZ there are three paths of G from v to Z{v}, mutually disjoint except for v. Moreover, we can find two “different” such subdivisions, if one exists.

Our proof is based on Mader's S-paths theorem.



中文翻译:

四个顶点上的根拓扑次要

对于一个图形ģ和一组ž的四个不同的顶点ģ,一个金刚石Ž是的子图G ^,使得对于一些标记Z={v1,v2,v3,v4}, 存在三个内部不相交的路径 1,2,3 带端点 v1,v2v3,v41,2, 分别。因此,这产生了一个4-- 在Z上使用分支顶点进行细分。

我们在四个顶点的规定集合Z上刻画不包含菱形的图G,假设对于每个vZvG有 3 条路径Z-{v}, 除v外互不相交。此外,我们可以找到两个“不同的”这样的细分,如果存在的话。

我们的证明基于 Mader 的S路径定理。

更新日期:2021-06-04
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