Csaba D. Tóth

The Szemerédi-Trotter theorem in the complex plane

It is shown that n points and e lines in the complex Euclidean plane ℂ2 determine O(n2/3e2/3 + n + e) point-line incidences. This bound is the best possible, and it generalizes the celebrated theorem by Szemerédi and Trotter about point-line incidences in the real Euclidean plane ℝ2.

Plane Geometric Graph Augmentation: A Generic Perspective

Graph augmentation problems are motivated by network design and have been studied extensively in optimization. We consider augmentation problems over plane geometric graphs, that is, graphs given with a crossing-free straight-line embedding in the plane. The geometric constraints on the possible new edges render some of the simplest augmentation problems intractable, and in many cases only extremal results are known. We survey recent results, highlight common trends, and gather numerous conjectures and open problems.

Connectivity augmentation in planar straight line graphs

It is shown that every connected planar straight line graph with n>=3 vertices has an embedding preserving augmentation to a 2-edge connected planar straight line graph with at most @?(2n-2)/3@? new edges. It is also shown that every planar straight line tree with n>=3 vertices has an embedding preserving augmentation to a 2-edge connected planar topological graph with at most @?n/2@? new edges. These bounds are the best possible. However, for every n>=3, there are planar straight line trees with n vertices that do not have an embedding preserving augmentation to a 2-edge connected planar straight line graph with fewer than 1733n-O(1) new edges.

Disjoint Compatible Geometric Matchings

We prove that for every set of n pairwise disjoint line segments in the plane in general position, where n is even, there is another set of n segments such that the 2n segments form pairwise disjoint simple polygons in the plane. This settles in the affirmative the Disjoint Compatible Matching Conjecture by Aichholzerxa0et al. (Comput. Geom. 42:617–626, 2009). The key tool in our proof is a novel subdivision of the free space around n disjoint line segments into at most n+1 convex cells such that the dual graph of the subdivision contains two edge-disjoint spanning trees.

Packing anchored rectangles

Let S be a set of n points in the unit square [0,1]2, one of which is the origin. We construct n pairwise interior-disjoint axis-aligned empty rectangles such that the lower left corner of each rectangle is a point in S, and the rectangles jointly cover at least a positive constant area (about 0.09). This is a first step towards the solution of a longstanding conjecture that the rectangles in such a packing can jointly cover an area of at least 1/2.

Vertex-Colored Encompassing Graphs

It is shown that every disconnected vertex-colored plane straight line graph with no isolated vertices can be augmented (by adding edges) into a connected plane straight line graph such that the new edges respect the coloring and the degree of every vertex increases by at most two. The upper bound for the increase of vertex degrees is best possible: there are input graphs that require the addition of two new edges incident to a vertex. The exclusion of isolated vertices is necessary: there are input graphs with isolated vertices that cannot be augmented to a connected vertex-colored plane straight line graph.

Constrained tri-connected planar straight line graphs ⁄

It is known that for any set V of n ≥ 4 points in the plane, not in convex position, there is a 3-connected planar straight line graph G = (V, E) with at most 2n − 2 edges, and this bound is the best possible. We show that the upper bound | E | ≤ 2n continues to hold if G = (V, E) is constrained to contain a given graph G 0 = (V, E 0), which is either a 1-factor (i.e., disjoint line segments) or a 2-factor (i.e., a collection of simple polygons), but no edge in E 0 is a proper diagonal of the convex hull of V. Since there are 1- and 2-factors with n vertices for which any 3-connected augmentation has at least 2n − 2 edges, our bound is nearly tight in these cases. We also examine possible conditions under which this bound may be improved, such as when G 0 is a collection of interior-disjoint convex polygons in a triangular container.

Open Guard Edges and Edge Guards in Simple Polygons

An open edge of a simple polygon is the set of points in the relative interior of an edge. We revisit several art gallery problems, previously considered for closed edge guards, using open edge guards. A guard edge of a polygon is an edge that sees every point inside the polygon. We show that every simple non-starshaped polygon admits at most one open guard edge, and give a simple new proof that it admits at most three closed guard edges. We also characterize open guard edges using a special type of kernel. Finally, we present lower bound constructions for simple polygons with n vertices that require (lfloor n/3 rfloor) open edge guards, and conjecture that this bound is tight.

Gap-Planar Graphs

We introduce the family of k-gap-planar graphs for (k ge 0), i.e., graphs that have a drawing in which each crossing is assigned to one of the two involved edges and each edge is assigned at most k of its crossings. This definition finds motivation in edge casing, as a (k)-gap-planar graph can be drawn crossing-free after introducing at most k local gaps per edge. We obtain results on the maximum density, drawability of complete graphs, complexity of the recognition problem, and relationships with other families of beyond-planar graphs.

Recognizing Weakly Simple Polygons

We present an