The Grid Theorem for vertex-minors

A survey of χ‐boundedness

“…Choi, Kwon, Oum, and Wollan [22] proved that every ideal closed under vertex‐minors not containing all wheels is ‐bounded. This was extended by a recent result of Geelen, Kwon, McCarty, and Wollan [57], proving the same for every ideal closed under vertex‐minors that does not include all circle graphs. These results are both superceded by a very recent result by James Davies, claiming that Conjecture 12.17 is true in general.…”

Stable graphs of bounded twin-width

“…For instance, the boundedness of treewidth and of cliquewidth delimits the area of algorithmic tractability of two natural variants of the monadic second-order logic (MSO) on graphs, in the sense of the existence of a fixed-parameter algorithm for model checking [Cou90,CMR00]. Further, both parameters admit duality theorems linking them to the largest size of a grid that can be embedded in the considered graph as a minor (for treewidth) or as a vertex minor (for cliquewidth) [RS86,GKMW20]. Finally, cliquewidth "projects" to treewidth once we restrict attention to sparse graph in the following sense: every class of graphs C that has bounded cliquewidth and is weakly sparse, in fact has bounded treewidth.…”

The Erdős-Hajnal Property for Graphs with No Fixed Cycle as a Pivot-Minor

“…Theorem 29 (Geelen, Kwon, McCarty, and Wollan [12]). For each circle graph H, there is an integer r(H) such that every graph with no vertex-minor isomorphic to H has rankwidth at most r(H).…”