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For a finite collection of graphs ${\cal F}$, the ${\cal F}$-M-DELETION (resp. ${\cal F}$-TM-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 (resp. topological minor). We are interested in the parameterized complexity of both problems when the parameter is the treewidth of $G$, denoted by $tw$, and specifically in the cases where ${\cal F}$ contains a single connected planar graph $H$. We present algorithms running in time $2^{O(tw)} \cdot n^{O(1)}$, called single-exponential, when $H$ is either $P_3$, $P_4$, $C_4$, the paw, the chair, and the banner for both $\{H\}$-M-DELETION and $\{H\}$-TM-DELETION, and when $H=K_{1,i}$, with $i \geq 1$, for $\{H\}$-TM-DELETION. Some of these algorithms use the rank-based approach introduced by Bodlaender et al. [Inform Comput, 2015]. This is the second 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 $H$., 36 pages, 2 figures

For a finite collection of graphs ${\cal F}$, the ${\cal F}$-M-DELETION problem consists in, given a graph $G$ and an integer $k$, deciding 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 prove that $f_{{\cal F}}(tw) = 2^{2^{O(tw \cdot\log tw)}}$ for every collection ${\cal F}$, that $f_{{\cal F}}(tw) = 2^{O(tw \cdot\log tw)}$ if ${\cal F}$ contains a planar graph, and that $f_{{\cal F}}(tw) = 2^{O(tw)}$ if in addition the input graph $G$ is planar or embedded in a surface. We also consider the version of the problem where the graphs in ${\cal F}$ are forbidden as topological minors, called ${\cal F}$-TM-DELETION. We prove similar results for this problem, except that in the last two algorithms, instead of requiring ${\cal F}$ to contain a planar graph, we need it to contain a subcubic planar graph. This is the first of a series of articles on this topic., Comment: 36 pages

International audience; For a fixed graph H, we are interested in the parameterized complexity of the following problem, called {H}-M-Deletion, parameterized by the treewidth tw of the input graph: given an n-vertex graph G and an integer k, decide whether there exists S ⊆ V (G) with |S| ≤ k such that G\S does not contain H as a minor. In previous work [IPEC, 2017] we proved that if H is planar and connected, then the problem cannot be solved in time 2 o(tw) · n O(1) under the ETH, and can be solved in time 2 O(tw·log tw) · n O(1). In this article we manage to classify the optimal asymptotic complexity of {H}-M-Deletion when H is a connected planar graph on at most 5 vertices. Out of the 29 possibilities (discarding the trivial case H = K 1), we prove that 9 of them are solvable in time 2 Θ(tw) · n O(1) , and that the other 20 ones are solvable in time 2 Θ(tw·log tw) · n O(1). Namely, we prove that K 4 and the diamond are the only graphs on at most 4 vertices for which the problem is solvable in time 2 Θ(tw·log tw) · n O(1) , and that the chair and the banner are the only graphs on 5 vertices for which the problem is solvable in time 2 Θ(tw) · n O(1). For the version of the problem where H is forbidden as a topological minor, the case H = K 1,4 can be solved in time 2 Θ(tw) · n O(1). This exhibits, to the best of our knowledge, the first difference between the computational complexity of both problems. 2012 ACM Subject Classification Mathematics of computing → Graph algorithms, Theory of computation → Parameterized complexity and exact algorithms

International audience; For a fixed collection of graphs F, the F-M-DELETION problem consists in, given a graph G and an integer k, decide whether there exists a subset S of V(G) of size at most 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. Using and enhancing the machinery of boundaried graphs and small sets of representatives introduced by Bodlaender et al. [J ACM, 2016], we prove that when all the graphs in F are connected and at least one of them is planar, then f_F(w) = 2^{O(wlog w)}. When F is a singleton containing a clique, a cycle, or a path on i vertices, we prove the following asymptotically tight bounds: - f_{K_4}(w) = 2^{Theta(wlog w)}. - f_{C_i}(w) = 2^{Theta(w)} for every i4. - f_{P_i}(w) = 2^{Theta(w)} for every i5. The lower bounds hold unless the Exponential Time Hypothesis fails, and the superexponential ones are inspired by a reduction of Marcin Pilipczuk [Discrete Appl Math, 2016]. The single-exponential algorithms use, in particular, the rank-based approach introduced by Bodlaender et al. [Inform Comput, 2015]. We also consider the version of the problem where the graphs in F are forbidden as topological minors, and prove essentially the same set of results holds.

International audience; Several min–max relations in graph theory can be expressed in the framework of the Erdős–Pósa property. Typically, this property reveals a connection between packing and covering problems on graphs. We describe some recent techniques for proving this property that are related to tree-like decompositions. We also provide an unified presentation of the current state of the art on this topic.

International audience; The Contraction Checking problem asks, given two graphs H and G as input, whether H can be obtained from G by a sequence of edge contractions. Contraction Checking remains NP-complete, even when H is fixed. We show that this is not the case when G is embeddable in a surface of fixed Euler genus. In particular, we give an algorithm that solves Contraction Checking in f(h,g)*|V(G)|^3 steps, where h is the size of H and g is the Euler genus of the input graph G.