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 , there are three internally disjoint paths with end vertices with on , respectively. Therefore, this yields a -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 there are three paths of G from v to , 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,一个金刚石上Ž是的子图G ^,使得对于一些标记, 存在三个内部不相交的路径 带端点 和 上 , 分别。因此,这产生了一个- 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,假设对于每个从v到G有 3 条路径, 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。