Notes on Graph Product Structure Theory
Notes on Graph Product Structure Theory Zdenekˇ Dvorák,ˇ Tony Huynh, Gwenaël Joret, Chun-Hung Liu, and David R. Wood Abstract 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 gener- alisations 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 prod- uct structure theorems for various geometrically defined graph classes, and present several open problems. Zdenekˇ Dvorákˇ Charles University, Prague, Czech Republic Supported by project 17-04611S (Ramsey-like aspects of graph coloring) of Czech Science Foun- dation e-mail: rakdver@iuuk.mff.cuni.cz Tony Huynh and David R. Wood School of Mathematics, Monash University, Melbourne, Australia Research supported by the Australian Research Council e-mail: tony.bourbaki@gmail.com,david.wood@monash.edu Gwenaël Joret Département d’Informatique, Université libre de Bruxelles, Brussels, Belgium Research supported by the Wallonia-Brussels Federation of Belgium, and by the Australian Re- search Council e-mail: gjoret@ulb.ac.be Chun-Hung Liu Department of Mathematics, Texas A&M University, College Station, Texas, USA Partially supported by the NSF under Grant No. DMS-1929851 e-mail: chliu@math.tamu.edu 1 2 Zdenekˇ Dvorák,ˇ Tony Huynh, Gwenaël Joret, Chun-Hung Liu, and David R.
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