SIAM Journal on Discrete Mathematics, Volume 34, Issue 1, Page 972-999, January 2020.
For a class $\mathcal C$ of graphs $G$ equipped with functions $f_G$ defined on subsets of $E(G)$ or $V(G)$, we say that $\mathcal{C}$ is $k$-$scattered$ with respect to $f_G$ if there exists a constant $\ell$ such that for every graph $G\in \mathcal C$, the domain of $f_G$ can be partitioned into subsets of size at most $k$ so that the union of every collection of the subsets has $f_G$ value at most $\ell$. We present structural characterizations of graph classes that are $k$-scattered with respect to several graph connectivity functions. In particular, our theorem for cut-rank functions provides a rough structural characterization of graphs having no $mK_{1,n}$ vertex-minor, which allows us to prove that such graphs have bounded linear rank-width.

中文翻译：

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图的分散类

SIAM离散数学杂志，第34卷，第1期，第972-999页，2020年1月。
对于配备了在$ E（G）$或$ V（G）$的子集中定义的函数$ f_G $的图$ G $的类$ \ mathcal C $，我们说$ \ mathcal {C} $为$ k相对于$ f_G $的$-$ scattered $，如果存在常量$ \ ell $，这样对于\ mathcal C $中的每个图形$ G \，$ f_G $的域最多可以划分为大小为$的子集k $，以便每个子集集合的并集最多具有$ f_G $值$ \ ell $。我们介绍了相对于几个图连接功能，$ k $分散的图类的结构特征。特别地，我们的秩函数的定理提供了没有$ mK_ {1，n} $顶点次幂的图的粗略结构特征，这使我们能够证明这种图具有线性秩宽度。