A fixed-parameter tractable algorithm for elimination distance to bounded degree graphs

In the literature on parameterized graph problems, there has been an increased effort in recent years aimed at exploring novel notions of graph edit-distance that are more powerful than the size of a modulator to a specific graph class. In this line of research, Bulian and Dawar [Algorithmica, 75 (2016), pp. 363--382] introduced the notion of elimination distance and showed that deciding whether a given graph has elimination distance at most $k$ to any minor-closed class of graphs is fixed-parameter tractable parameterized by $k$ [Algorithmica, 79 (2017), pp. 139--158]. They showed that graph isomorphism parameterized by the elimination distance to bounded degree graphs is fixed-parameter tractable and asked whether determining the elimination distance to the class of bounded degree graphs is fixed-parameter tractable. Recently, Lindermayr, Siebertz, and Vigny [MFCS 2020, LIPIcs Leibniz Int. Proc. Inform. 170, Wadern Germany, 2020, 65] obtained a fixed-parameter algorithm for this problem in the special case where the input is restricted to $K_5$-minor free graphs. In this paper, we answer the question of Bulian and Dawar in the affirmative for general graphs. In fact, we give a more general result capturing elimination distance to any graph class characterized by a finite set of graphs as forbidden induced subgraphs