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.

Keywords

  1. graph structure
  2. vertex-minor
  3. subgraph

MSC codes

  1. 05C75

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Information & Authors

Information

Published In

cover image SIAM Journal on Discrete Mathematics
SIAM Journal on Discrete Mathematics
Pages: 972 - 999
ISSN (online): 1095-7146

History

Submitted: 16 October 2019
Accepted: 20 February 2020
Published online: 31 March 2020

Keywords

  1. graph structure
  2. vertex-minor
  3. subgraph

MSC codes

  1. 05C75

Authors

Affiliations

Funding Information

Horizon 2020 Framework Programme https://doi.org/10.13039/100010661 : 648527
Institute for Basic Science https://doi.org/10.13039/501100010446 : IBS-R029-C1
National Research Foundation of Korea https://doi.org/10.13039/501100003725 : NRF-2018R1D1A1B07050294, NRF-2017R1A2B4005020

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