Deleting, Eliminating and Decomposing to Hereditary Classes Are All FPT-Equivalent
For a graph class H, the graph parameters elimination distance to H (denoted by ed_ H) [Bulian and Dawar, Algorithmica, 2016], and H-treewidth (denoted by tw_ H) [Eiben et al. JCSS, 2021] aim to minimize the treedepth and treewidth, respectively, of the "torso" of the graph induced on a modulator to the graph class H. Here, the torso of a vertex set S in a graph G is the graph with vertex set S and an edge between two vertices u, v ∈ S if there is a path between u and v in G whose internal
... ces all lie outside S. In this paper, we show that from the perspective of (non-uniform) fixed-parameter tractability (FPT), the three parameters described above give equally powerful parameterizations for every hereditary graph class H that satisfies mild additional conditions. In fact, we show that for every hereditary graph class H satisfying mild additional conditions, with the exception of tw_ H parameterized by ed_ H, for every pair of these parameters, computing one parameterized by itself or any of the others is FPT-equivalent to the standard vertex-deletion (to H) problem. As an example, we prove that an FPT algorithm for the vertex-deletion problem implies a non-uniform FPT algorithm for computing ed_ H and tw_ H. The conclusions of non-uniform FPT algorithms being somewhat unsatisfactory, we essentially prove that if H is hereditary, union-closed, CMSO-definable, and (a) the canonical equivalence relation (or any refinement thereof) for membership in the class can be efficiently computed, or (b) the class admits a "strong irrelevant vertex rule", then there exists a uniform FPT algorithm for ed_ H.