Jean-Florent Raymond

Low Polynomial Exclusion of Planar Graph Patterns

The celebrated grid exclusion theorem states that for every h-vertex planar graph H, there is a constant c h such that if a graph G does not contain H as a minor then G has treewidth at most c h. We are looking for patterns of H where this bound can become a low degree polynomial. We provide such bounds for the following parameterized graphs: the wheel (c h = O(h)), the double wheel (c h = O(h 2 · log 2 h)), any graph of pathwidth at most 2 (c h = O(h 2)), and the yurt graph (c h = O(h 4)).

Cutwidth: obstructions and algorithmic aspects

Cutwidth is one of the classic layout parameters for graphs. It measures how well one can order the vertices of a graph in a linear manner, so that the maximum number of edges between any prefix and its complement suffix is minimized. As graphs of cutwidth at most k are closed under taking immersions, the results of Robertson and Seymour imply that there is a finite list of minimal immersion obstructions for admitting a cut layout of width at most k. We prove that every minimal immersion obstruction for cutwidth at most k has size at most

Minors in graphs of large θr-girth

2^{{O}(k^3\log k)}

Induced minors and well-quasi-ordering

2O(k3logk). As an interesting algorithmic byproduct, we design a new fixed-parameter algorithm for computing the cutwidth of a graph that runs in time

Well-quasi-ordering H-contraction-free graphs ∗

2^{{O}(k^2\log k)}\cdot n

Scattered packings of cycles

2O(k2logk)·n, where k is the optimum width and n is the number of vertices. While being slower by a