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- W4287272862 abstract "For a finite collection of graphs ${cal F}$, the ${cal F}$-M-DELETION problem consists in, given a graph $G$ and an integer $k$, decide whether there exists $S subseteq V(G)$ with $|S| leq k$ such that $G setminus S$ does not contain any of the graphs in ${cal F}$ as a minor. We are interested in the parameterized complexity of ${cal F}$-M-DELETION when the parameter is the treewidth of $G$, denoted by $tw$. Our objective is to determine, for a fixed ${cal F}$, the smallest function $f_{{cal F}}$ such that ${cal F}$-M-DELETION can be solved in time $f_{{cal F}}(tw) cdot n^{O(1)}$ on $n$-vertex graphs. We provide lower bounds under the ETH on $f_{{cal F}}$ for several collections ${cal F}$. We first prove that for any ${cal F}$ containing connected graphs of size at least two, $f_{{cal F}}(tw)= 2^{Omega(tw)}$, even if the input graph $G$ is planar. Our main contribution consists of superexponential lower bounds for a number of collections ${cal F}$, inspired by a reduction of Bonnet et al.~[IPEC, 2017]. In particular, we prove that when ${cal F}$ 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 $f_{{cal F}}(tw)= 2^{Omega(tw cdot log tw)}$. This is the third of a series of articles on this topic, and the results given here together with other ones allow us, in particular, to provide a tight dichotomy on the complexity of ${H}$-M-DELETION, in terms of $H$, when $H$ is connected." @default.
- W4287272862 created "2022-07-25" @default.
- W4287272862 creator A5025941213 @default.
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- W4287272862 date "2021-03-11" @default.
- W4287272862 modified "2023-09-23" @default.
- W4287272862 title "Hitting minors on bounded treewidth graphs. III. Lower bounds" @default.
- W4287272862 doi "https://doi.org/10.48550/arxiv.2103.06614" @default.
- W4287272862 hasPublicationYear "2021" @default.
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