Stefan Felsner

Convex drawings of planar graphs and the order dimension of 3-polytopes

We define an analogue of Schnyders tree decompositions for 3-connected planar graphs. Based on this structure we obtain:• Let G be a 3-connected planar graph with f faces, then G has a convex drawing with its vertices embedded on the (f−1)×(f−1) grid.• Let G be a 3-connected planar graph. The dimension of the incidence order of vertices, edges and bounded faces of G is at most 3.The second result is originally due to Brightwell and Trotter. Here we give a substantially simpler proof.

Trapezoid graphs and generalizations, geometry and algorithms

Trapezoid graphs are a class of cocomparability graphs containing interval graphs and permutation graphs as subclasses. They were introduced by Dagan, Golumbic and Pinter [DGP]. They propose an O(n2) algorithm for chromatic number and a less efficient algorithm for maximum clique on trapezoid graphs. Based on a geometric representation of trapezoid graphs by boxes in the plane we design optimal, i.e., O(n log n), algorithms for chromatic number, weighted independent set, clique cover and maximum weighted clique on such graphs. We generalize trapezoid graphs to so called k-trapezoidal graphs. The ideas behind the clique cover and weighted independent set algorithms for trapezoid graphs carry over to higher dimensions. This leads to O(n logk−1n) algorithms for k-trapezoidal graphs. We also propose a new class of graphs called circle trapezoid graphs. This class contains trapezoid graphs, circle graphs and circular-arc graphs as subclasses. We show that clique and independent set problems for circle trapezoid graphs are still polynomially solvable. The algorithms solving these two problems require algorithms for trapezoid graphs as subroutines.

Convex Drawings of 3-Connected Plane Graphs

AbstractWe use Schnyder woods of 3-connected planar graphs to produce convex straight-line drawings on a grid of size

Recognition Algorithms for Orders of Small Width and Graphs of Small Dilworth Number

(n-2-\Delta)\times (n-2-\Delta).

Geodesic Embeddings and Planar Graphs

The parameter

Schnyder Woods and Orthogonal Surfaces

\Delta\geq 0

Rectangle and Square Representations of Planar Graphs

depends on the Schnyder wood used for the drawing. This parameter is in the range

On-line chain partitions of orders

0 \leq \Delta\leq {n}/{2}-2.

On the Number of Arrangements of Pseudolines

The algorithm is a refinement of the face-counting algorithm; thus, in particular, the size of the grid is at most

Dimension, graph and hypergraph coloring

(f-2)\times(f-2).