3 research outputs found
Graph product structure for non-minor-closed classes
Dujmovi\'c et al. (FOCS 2019) recently proved that every planar graph is a
subgraph of the strong product of a graph of bounded treewidth and a path.
Analogous results were obtained for graphs of bounded Euler genus or
apex-minor-free graphs. These tools have been used to solve longstanding
problems on queue layouts, non-repetitive colouring, -centered colouring,
and adjacency labelling. This paper proves analogous product structure theorems
for various non-minor-closed classes. One noteable example is -planar graphs
(those with a drawing in the plane in which each edge is involved in at most
crossings). We prove that every -planar graph is a subgraph of the
strong product of a graph of treewidth and a path. This is the first
result of this type for a non-minor-closed class of graphs. It implies, amongst
other results, that -planar graphs have non-repetitive chromatic number
upper-bounded by a function of . All these results generalise for drawings
of graphs on arbitrary surfaces. In fact, we work in a much more general
setting based on so-called shortcut systems that are of independent interest.
This leads to analogous results for map graphs, string graphs, graph powers,
and nearest neighbour graphs.Comment: v2 Cosmetic improvements and a corrected bound for
(layered-)(tree)width in Theorems 2, 9, 11, and Corollaries 1, 3, 4, 6, 12.
v3 Complete restructur
On Crossing Numbers of Geometric Proximity Graphs
Let P be a set of n points in the plane. A geometric proximity graph on P is a graph where two points are connected by a straight-line segment if they satisfy some prescribed proximity rule. We consider four classes of higher order proximity graphs, namely, the k-nearest neighbor graph, the k-relative neighborhood graph, the k-Gabriel graph and the k-Delaunay graph. For k = 0 (k = 1 in the case of the k-nearest neighbor graph) these graphs are plane, but for higher values of k in general they contain crossings. In this paper we provide lower and upper bounds on their minimum and maximum number of crossings. We give general bounds and we also study particular cases that are especially interesting from the viewpoint of applications. These cases include the 1-Delaunay graph and the k-nearest neighbor graph for small values of k