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Hitting minors on bounded treewidth graphs. III. Lower bounds

Julien Baste 1 Ignasi Sau Valls 1 Dimitrios M. Thilikos 1
1 ALGCO - Algorithmes, Graphes et Combinatoire
LIRMM - Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier
Abstract : For a finite collection of graphs F, the F-M-Deletion problem consists in, given a graph G and an integer k, decide whether there exists S ⊆ V (G) with |S| ≤ k such that G \ S does not contain any of the graphs in F as a minor. We are interested in the parameterized complexity of F-M-Deletion when the parameter is the treewidth of G, denoted by tw. Our objective is to determine, for a fixed F, the smallest function f F such that F-M-Deletion can be solved in time f F (tw) · n O(1) on n-vertex graphs. We provide lower bounds under the ETH on f F for several collections F. We first prove that for any F containing connected graphs of size at least two, f F (tw) = 2 Ω(tw) , even if the input graph G is planar. Our main contribution consists of superexponential lower bounds for a number of collections F, inspired by a reduction of Bonnet et al. [IPEC, 2017]. In particular, we prove that when F contains a single connected graph H that is either P 5 or is not a minor of the banner (that is, the graph consisting of a C 4 plus a pendent edge), then f F (tw) = 2 Ω(tw·log tw). This is the third of a series of articles on this topic, and the lower bounds given here together with the algorithms given in the first two articles yield a dichotomy on the complexity of {H}-M-Deletion in terms of H, when H is a connected planar graph with |V (H)| ≥ 2. Namely, f {H} (tw) = 2 Θ(tw) if H is a minor of the banner that is different from P 5 , and f {H} (tw) = 2 Θ(tw·log tw) otherwise.
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Submitted on : Thursday, November 5, 2020 - 12:53:41 PM
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Julien Baste, Ignasi Sau Valls, Dimitrios M. Thilikos. Hitting minors on bounded treewidth graphs. III. Lower bounds. Journal of Computer and System Sciences, Elsevier, 2020, 109, pp.56-77. ⟨10.1016/j.jcss.2019.11.002⟩. ⟨lirmm-02989938⟩



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