Chudnovsky, MariaScott, AlexSeymour, PaulSpirkl, Sophie2023-11-212023-11-212023-11https://doi.org/10.1016/j.jctb.2023.07.004http://hdl.handle.net/10012/20111© 2023 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).What can be said about the structure of graphs that do not contain an induced copy of some graph H? Rödl showed in the 1980s that every H-free graph has large parts that are very sparse or very dense. More precisely, let us say that a graph F on n vertices is ε-restricted if either F or its complement has maximum degree at most εn. Rödl proved that for every graph H, and every ε > 0, every H-free graph G has a linear-sized set of vertices inducing an ε-restricted graph. We strengthen Rödl’s result as follows: for every graph H, and all ε > 0, every H-free graph can be partitioned into a bounded number of subsets inducing ε-restricted graphs.enAttribution-NonCommercial-NoDerivatives 4.0 Internationalhttp://creativecommons.org/licenses/by-nc-nd/4.0/induced subgraphssparse graphsArticle