论文标题
弹性元素的分离器定理，$ 3 $连接的Matroids
A splitter theorem for elastic elements in $3$-connected matroids
论文作者
论文摘要
如果$ {\ rm si}（m/e）$，$ 3 $连接的Matroid $ m $的元素$ e $是弹性的，简化了$ m/e $，以及$ {\ rm co}（m \ rm co}（m \ backslash e）$，$ m \ m \ backslash e $ cosimplification of $ m \ backslash e $，是$ 3 $ -bacnect。最近显示的是，如果$ | e（m）| \ geq 4 $，那么$ m $至少有四个弹性元素，因为$ m $没有$ 4 $ element粉丝，也没有特定家族的$ 3 $ separators的成员。在本文中，我们将此车轮和旋转类型的结果扩展到了分离器定理，在该定理中，删除元素是关于弹性的，并保留指定的$ 3 $连接的未成年人。我们还证明，如果$ m $具有四个弹性元素，那么它具有三个路径宽度。最后，我们解决了一个惠特尔和威廉姆斯的问题，并表明过去的类似结果（去除元素相对于固定基础）是这项工作的后果。
An element $e$ of a $3$-connected matroid $M$ is elastic if ${\rm si}(M/e)$, the simplification of $M/e$, and ${\rm co}(M\backslash e)$, the cosimplification of $M\backslash e$, are both $3$-connected. It was recently shown that if $|E(M)|\geq 4$, then $M$ has at least four elastic elements provided $M$ has no $4$-element fans and no member of a specific family of $3$-separators. In this paper, we extend this wheels-and-whirls type result to a splitter theorem, where the removal of elements is with respect to elasticity and keeping a specified $3$-connected minor. We also prove that if $M$ has exactly four elastic elements, then it has path-width three. Lastly, we resolve a question of Whittle and Williams, and show that past analogous results, where the removal of elements is relative to a fixed basis, are consequences of this work.
