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For a finite collection of graphs F , the F -M-Deletion (resp. F -TM-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 (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 F contains a single connected planar graph H. We present algorithms running in time 2 O (tw) ⋅ 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 ≥ 1 , for { H } -TM-Deletion. Some of these algorithms use the rank-based approach introduced by Bodlaender et al. (2015) [7]. 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. [ABSTRACT FROM AUTHOR]

For some years it was believed that for “connectivity” problems such as Hamiltonian Cycle , algorithms running in time 2 O ( tw ) ⋅ n O ( 1 ) – called single-exponential – existed only on planar and other topologically constrained graph classes, where tw stands for the treewidth of the n -vertex input graph. This was recently disproved by Cygan et al. [ 3 , FOCS 2011], Bodlaender et al. [ 1 , ICALP 2013], and Fomin et al. [ 11 , SODA 2014], who provided single-exponential algorithms on general graphs for most connectivity problems that were known to be solvable in single-exponential time on topologically constrained graphs. In this article we further investigate the role of planarity in connectivity problems parameterized by treewidth, and convey that several problems can indeed be distinguished according to their behavior on planar graphs. Known results from the literature imply that there exist problems, like Cycle Packing , that cannot be solved in time 2 o ( tw log ⁡ tw ) ⋅ n O ( 1 ) on general graphs but that can be solved in time 2 O ( tw ) ⋅ n O ( 1 ) when restricted to planar graphs. Our main contribution is to show that there exist natural problems that can be solved in time 2 O ( tw log ⁡ tw ) ⋅ n O ( 1 ) on general graphs but that cannot be solved in time 2 o ( tw log ⁡ tw ) ⋅ n O ( 1 ) even when restricted to planar graphs. Furthermore, we prove that Planar Cycle Packing and Planar Disjoint Paths cannot be solved in time 2 o ( tw ) ⋅ n O ( 1 ) . The mentioned negative results hold unless the ETH fails. We feel that our results constitute a first step in a subject that can be further exploited. [ABSTRACT FROM AUTHOR]

In an edge-colored graph, the cost incurred at a vertex on a path when two incident edges with different colors are traversed is called reload or changeover cost. The Minimum Changeover Cost Arborescence ( MinCCA ) problem consists in finding an arborescence with a given root vertex such that the total changeover cost of the internal vertices is minimized. It has been recently proved by Gözüpek et al. (2016) that the MinCCA problem when parameterized by the treewidth and the maximum degree of the input graph is in FPT . In this article we present the following hardness results for MinCCA : • the problem is W [1]-hard when parameterized by the vertex cover number of the input graph, even on graphs of degeneracy at most 3. In particular, it is W [1]-hard parameterized by the treewidth of the input graph, which answers the main open problem in the work of Gözüpek et al. (2016); • it is W [1]-hard on multigraphs parameterized by the tree-cutwidth of the input multigraph; and • it remains NP -hard on planar graphs even when restricted to instances with at most 6 colors and 0/1 symmetric costs, or when restricted to instances with at most 8 colors, maximum degree bounded by 4, and 0/1 symmetric costs. [ABSTRACT FROM AUTHOR]