Tree pivotminors and linear rankwidth
Abstract
Treewidth and its linear variant pathwidth play a central role for the graph minor relation. In particular, Robertson and Seymour (1983) proved that for every tree~$T$, the class of graphs that do not contain $T$ as a minor has bounded pathwidth. For the pivotminor relation, rankwidth and linear rankwidth take over the role from treewidth and pathwidth. As such, it is natural to examine if for every tree~$T$, the class of graphs that do not contain $T$ as a pivotminor has bounded linear rankwidth. We first prove that this statement is false whenever $T$ is a tree that is not a caterpillar. We conjecture that the statement is true if $T$ is a caterpillar. We are also able to give partial confirmation of this conjecture by proving: (1) for every tree $T$, the class of $T$pivotminorfree distancehereditary graphs has bounded linear rankwidth if and only if $T$ is a caterpillar; (2) for every caterpillar $T$ on at most four vertices, the class of $T$pivotminorfree graphs has bounded linear rankwidth. To prove our second result, we only need to consider $T=P_4$ and $T=K_{1,3}$, but we follow a general strategy: first we show that the class of $T$pivotminorfree graphs is contained in some class of $(H_1,H_2)$free graphs, which we then show to have bounded linear rankwidth. In particular, we prove that the class of $(K_3,S_{1,2,2})$free graphs has bounded linear rankwidth, which strengthens a known result that this graph class has bounded rankwidth.
 Publication:

arXiv eprints
 Pub Date:
 August 2020
 arXiv:
 arXiv:2008.00561
 Bibcode:
 2020arXiv200800561D
 Keywords:

 Mathematics  Combinatorics;
 Computer Science  Discrete Mathematics
 EPrint:
 26 pages, 5 figures