Scattered classes of graphs
O-joung Kwon, Sang-il Oum

TL;DR
This paper introduces the concept of $k$-scattered classes of graphs based on functions like cut-rank, providing structural characterizations and implications for graphs excluding certain vertex-minors, such as bounded linear rank-width.
Contribution
It offers new structural characterizations of $k$-scattered graph classes with respect to various connectivity functions, including a characterization related to cut-rank functions.
Findings
Characterization of $k$-scattered classes for several connectivity functions
Structural insight into graphs excluding $mK_{1,n}$ vertex-minors
Graphs excluding certain vertex-minors have bounded linear rank-width
Abstract
For a class of graphs equipped with functions defined on subsets of or , we say that is -scattered with respect to if there exists a constant such that for every graph , the domain of can be partitioned into subsets of size at most so that the union of every collection of the subsets has value at most . We present structural characterizations of graph classes that are -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 vertex-minor, which allows us to prove that such graphs have bounded linear rank-width.
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