Flip Distance Between Triangulations of a Planar Point Set is APX-Hard

In this work we consider triangulations of point sets in the Euclidean plane, i.e., maximal straight-line crossing-free graphs on a finite set of points. Given a triangulation of a point set, an edge flip is the operation of removing one edge and adding another one, such that the resulting graph is again a triangulation. Flips are a major way of locally transforming triangular meshes. We show that, given a point set $S$ in the Euclidean plane and two triangulations $T_1$ and $T_2$ of $S$, it is an APX-hard problem to minimize the number of edge flips to transform $T_1$ to $T_2$.Comment: A previous version only showed NP-completeness of the corresponding decision problem. The current version is the one of the accepted manuscrip

Gabriel Triangulations and Angle-Monotone Graphs: Local Routing and Recognition

A geometric graph is angle-monotone if every pair of vertices has a path between them that---after some rotation---is $x$- and $y$-monotone. Angle-monotone graphs are $\sqrt 2$-spanners and they are increasing-chord graphs. Dehkordi, Frati, and Gudmundsson introduced angle-monotone graphs in 2014 and proved that Gabriel triangulations are angle-monotone graphs. We give a polynomial time algorithm to recognize angle-monotone geometric graphs. We prove that every point set has a plane geometric graph that is generalized angle-monotone---specifically, we prove that the half-$\theta_6$-graph is generalized angle-monotone. We give a local routing algorithm for Gabriel triangulations that finds a path from any vertex $s$ to any vertex $t$ whose length is within $1 + \sqrt 2$ times the Euclidean distance from $s$ to $t$. Finally, we prove some lower bounds and limits on local routing algorithms on Gabriel triangulations.Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016

Upward Book Embeddings of st-Graphs

We study k-page upward book embeddings (kUBEs) of st-graphs, that is, book embeddings of single-source single-sink directed acyclic graphs on k pages with the additional requirement that the vertices of the graph appear in a topological ordering along the spine of the book. We show that testing whether a graph admits a kUBE is NP-complete for k >= 3. A hardness result for this problem was previously known only for k = 6 [Heath and Pemmaraju, 1999]. Motivated by this negative result, we focus our attention on k=2. On the algorithmic side, we present polynomial-time algorithms for testing the existence of 2UBEs of planar st-graphs with branchwidth b and of plane st-graphs whose faces have a special structure. These algorithms run in O(f(b)* n+n^3) time and O(n) time, respectively, where f is a singly-exponential function on b. Moreover, on the combinatorial side, we present two notable families of plane st-graphs that always admit an embedding-preserving 2UBE

Competitive online routing in geometric graphs

AbstractWe consider online routing algorithms for finding paths between the vertices of plane graphs. Although it has been shown in Bose et al. (Internat. J. Comput. Geom. 12(4) (2002) 283) that there exists no competitive routing scheme that works on all triangulations, we show that there exists a simple online O(1)-memory c-competitive routing strategy that approximates the shortest path in triangulations possessing the diamond property, i.e., the total distance travelled by the algorithm to route a message between two vertices is at most a constant c times the shortest path. Our results imply a competitive routing strategy for certain classical triangulations such as the Delaunay, greedy, or minimum-weight triangulation, since they all possess the diamond property. We then generalize our results to show that the O(1)-memory c-competitive routing strategy works for all plane graphs possessing both the diamond property and the good convex polygon property