Scattered classes of graphs

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 $\mathcal C$ 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.