It was recently proved that every planar graph is a subgraph of the strong product of a path and a graph with bounded treewidth. This paper surveys generalisations of this result for graphs on surfaces, minor-closed classes, various non minor-closed classes, and graph classes with polynomial growth. We then explore how graph product structure might be applicable to more broadly defined graph classes. In particular, we characterise when a graph class defined by a cartesian or strong product has bounded or polynomial expansion. We then explore graph product structure theorems for various geometrically defined graph classes, and present several open problems.
|Title of host publication||2019-20 MATRIX Annals|
|Editors||David R Wood, Jan de Gier, Cheryl E Praeger, Terence Tao|
|Place of Publication||Cham Switzerland|
|Number of pages||21|
|ISBN (Print)||9783030624965, 9783030624996|
|Publication status||Published - 2021|
|Name||MATRIX Book Series|