Therese C. Biedl

Efficient algorithms for Petersen's matching theorem

Petersens theorem is a classic result in matching theory from 1891, stating that every 3-regular bridgeless graph has a perfect matching. Our work explores efficient algorithms for finding perfect matchings in such graphs. Previously, the only relevant matching algorithms were for general graphs, and the fastest algorithm ran in O(n3/2) time for 3-regular graphs. We have developed an O(nlog4n)-time algorithm for perfect matching in a 3-regular bridgeless graph. When the graph is also planar, we have as the main result of our paper an optimal O(n)-time algorithm. We present three applications of this result: terrain guarding, adaptive mesh refinement, and quadrangulation.

Curvature-Constrained Shortest Paths in a Convex Polygon

Let B be a point robot moving in the plane, whose path is constrained to have curvature at most 1, and let

Area-Efficient Static and Incremental Graph Drawings

\poly

Tight bounds on maximal and maximum matchings

be a convex polygon with n vertices. We study the collision-free, optimal path-planning problem for B moving between two configurations inside

On triangulating planar graphs under the four-connectivity constraint

\poly

The Three-Phase Method: A Unified Approach to Orthogonal Graph Drawing

. (A configuration specifies both a location and a direction of travel.) We present an O(n2 log n) time algorithm for determining whether a collision-free path exists for B between two given configurations. If such a path exists, the algorithm returns a shortest one. We provide a detailed classification of curvature-constrained shortest paths inside a convex polygon and prove several properties of them, which are interesting in their own right. For example, we prove that any such shortest path is comprised of at most eight segments, each of which is a circular arc of unit radius or a straight-line segment. Some of the properties are quite general and shed some light on curvature-constrained shortest paths amid obstacles.

A Better Heuristic for Orthogonal Graph Drawings

In this paper, we present algorithms to produce orthogonal drawings of arbitrary graphs. As opposed to most known algorithms, we do not restrict ourselves to graphs with maximum degree 4. The best previous result gave an \((m - 1) \times \left( {\tfrac{m}{2} + 1} \right)\)-grid for graphs with n nodes and m edges.

Balanced vertex-orderings of graphs

In this paper, we study lower bounds on the size of maximal and maximum matchings in 3-connected planar graphs and graphs with bounded maximum degree. For each class, we give a lower bound on the size of matchings, and show that the bound is tight for some graph within the class.

Drawing Planar Partitions II: HH-Drawings

Abstract. Triangulation of planar graphs under constraints is a fundamental problem in the representation of objects. Related keywords are graph augmentation from the field of graph algorithms and mesh generation from the field of computational geometry. We consider the triangulation problem for planar graphs under the constraint to satisfy 4-connectivity. A 4-connected planar graph has no separating triangles, i.e., cycles of length 3 which are not a face. We show that triangulating embedded planar graphs without introducing new separating triangles can be solved in linear time and space. If the initial graph had no separating triangle, the resulting triangulation is 4-connected. If the planar graph is not embedded, then deciding whether there exists an embedding with at most k separating triangles is NP-complete. For biconnected graphs a linear-time approximation which produces an embedding with at most twice the optimal number is presented. With this algorithm we can check in linear time whether a biconnected planar graph can be made 4-connected while maintaining planarity. Several related remarks and results are included.

A note on reconfiguring tree linkages: trees can lock

In this paper, we study orthogonal graph drawings from a practical point of view. Most previously existing algorithms restricted the attention to graphs of maximum degree four. Here we study orthogonal drawing algorithms that work for any input graph, and discuss different models for such drawings. Then we introduce the three-phase method, a generic technique to create high-degree orthogonal drawings. This approach simplifies the description and implementation of orthogonal graph drawing, and can be applied to global as well as interactive and incremental settings.