A k-ary semi-algebraic relation E on R-d is a subset of R-kd, the set of k-tuples of points in R-d, which is determined by a finite number of polynomial inequalities in kd real variables. The description complexity of such a relation is at most t if d, k < ...
The problem of finding "small" sets that meet every straight-line which intersects a given convex region was initiated by Mazurkiewicz in 1916. We call such a set an opaque set or a barrier for that region. We consider the problem of computing the shortest ...
We show that the lines of every arrangement of n lines in the plane can be colored with O(root n/log n) colors such that no face of the arrangement is monochromatic. This improves a bound of Bose et al. by a circle minus(root/log n) factor. Any further imp ...
Given d + 1 hyperplanes h(1),..., h(d+1) in general position in R-d, let Delta(h(1),..., h(d+1)) denote the unique bounded simplex enclosed by them. There exists a constant c(d) > 0 such that for any finite families H-1,..., Hd+1 of hyperplanes in R-d, the ...
A topological graph G is a graph drawn in the plane with vertices represented by points and edges represented by continuous arcs connecting the vertices. If every edge is drawn as a straight-line segment, then G is called a geometric graph. A k-grid in a t ...
Given a collection of n opaque unit disks in the plane, we want to find a stacking order for them that maximizes their visible perimeter, the total length of all pieces of their boundaries visible from above. We prove that if the centers of the disks form ...
A graph drawn in the plane is called k-quasi-planar if it does not contain k pair-wise crossing edges. It has been conjectured for a long time that for every fixed k, the maximum number of edges of a k-quasi-planar graph with n vertices is O(n). The best k ...
The graph obtained from the integer grid Z x Z by the removal of all horizontal edges that do not belong to the x-axis is called a comb. In a random walk on a graph, whenever a walker is at a vertex v, in the next step it will visit one of the neighbors of ...
Improving a result of Aichholzer et al., we show that there exists a constant c > 0 satisfying the following condition. Any two-colored set of n points in general position in the plane has at least cn(4/3) triples of the same color such that the triangles ...
Given a collection of n opaque unit disks in the plane, we want to find a stacking order for them that maximizes their visible perimeter, the total length of all pieces of their boundaries visible from above. We prove that if the centers of the disks form ...
In a seminal paper published in 1946, Erd ̋os initiated the investigation of the distribution of distances generated by point sets in metric spaces. In spite of some spectacular par- tial successes and persistent attacks by generations of mathe- maticians, ...
We settle a problem of Dujmovic, Eppstein, Suderman, and Wood by showing that there exists a function f with the property that every planar graph G with maximum degree d admits a drawing with noncrossing straight-line edges, using at most f(d) different sl ...
Erd\H{o}s conjectured in 1946 that every n-point set P in convex position in the plane contains a point that determines at least floor(n/2) distinct distances to the other points of P. The best known lower bound due to Dumitrescu (2006) is 13n/36 - O(1). I ...
Given a graph G, an obstacle representation of G is a set of points in the plane representing the vertices of G, together with a set of connected obstacles such that two vertices of G are joined by an edge if and only if the corresponding points can be con ...
Given a collection C of curves in the plane, its string graph is defined as the graph with vertex set C, in which two curves in C are adjacent if and only if they intersect. Given a partially ordered set (P,
