# A complexity dichotomy for hitting connected minors on bounded treewidth graphs: the chair and the banner draw the boundary

For a fixed connected graph H, the {H}-M-DELETION problem asks, given a graph G, for the minimum number of vertices that intersect all minor models of H in G. It is known that this problem can be solved in time f(tw)· n^O(1), where tw is the treewidth of G. We determine the asymptotically optimal function f(tw), for each possible choice of H. Namely, we prove that, under the ETH, f(tw) =2^Θ(tw) if H is a contraction of the chair or the banner, and f(tw)=2^Θ( tw ·log tw) otherwise. Prior to this work, such a complete characterization was only known when H is a planar graph with at most five vertices. For the upper bounds, we present an algorithm in time 2^Θ( tw·log tw)· n^O(1) for the more general problem where all minor models of connected graphs in a finite family F need to be hit. We combine several ingredients such as the machinery of boundaried graphs in dynamic programming via representatives, the Flat Wall Theorem, Bidimensionality, the irrelevant vertex technique, treewidth modulators, and protrusion replacement. In particular, this algorithm vastly generalizes a result of Jansen et al. [SODA 2014] for the particular case F={K_5,K_3,3}. For the lower bounds, our reductions are based on a generic construction building on the one given by the authors in [IPEC 2018], which uses the framework introduced by Lokshtanov et al. [SODA 2011] to obtain superexponential lower bounds.

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