János Pach

How to draw a planar graph on a grid

Answering a question of Rosenstiehl and Tarjan, we show that every plane graph withn vertices has a Fáry embedding (i.e., straight-line embedding) on the 2n−4 byn−2 grid and provide anO(n) space,O(n logn) time algorithm to effect this embedding. The grid size is asymptotically optimal and it had been previously unknown whether one can always find a polynomial sized grid to support such an embedding. On the other hand we show that any setF, which can support a Fáry embedding of every planar graph of sizen, has cardinality at leastn+(1−o(1))√n which settles a problem of Mohar.

On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles

Let γ1,..., γm bem simple Jordan curves in the plane, and letK1,...,Km be their respective interior regions. It is shown that if each pair of curves γi, γj,i ≠j, intersect one another in at most two points, then the boundary ofK=∩i=1mKi contains at most max(2,6m − 12) intersection points of the curvesγ1, and this bound cannot be improved. As a corollary, we obtain a similar upper bound for the number of points of local nonconvexity in the union ofm Minkowski sums of planar convex sets. Following a basic approach suggested by Lozano Perez and Wesley, this can be applied to planning a collision-free translational motion of a convex polygonB amidst several (convex) polygonal obstaclesA1,...,Am. Assuming that the number of corners ofB is fixed, the algorithm presented here runs in timeO (n log2n), wheren is the total number of corners of theAis.

Graphs drawn with few crossings per edge

We show that if a graph ofv vertices can be drawn in the plane so that every edge crosses at mostk>0 others, then its number of edges cannot exceed 4.108√kv. Fork≤4, we establish a better bound, (k+3)(v−2), which is tight fork=1 and 2. We apply these estimates to improve a result of Ajtai et al. and Leighton, providing a general lower bound for the crossing number of a graph in terms of its number of vertices and edges.

Fat Triangles Determine Linearly Many Holes

The authors show that for every fixed

Small sets supporting fary embeddings of planar graphs

\delta>0

Embedding planar graphs at fixed vertex locations

the following holds: If

Arrangements of curves in the plane—topology, combinatorics, and algorithms

F

Embedding a planar triangulation with vertices at specified points

is a union of

On the Number of Incidences Between Points and Curves

n

Almost tight bounds for e-nets

triangles, all of whose angles are at least