On Monotone Sequences of Directed Flips, Triangulations of Polyhedra, and Structural Properties of a Directed Flip Graph

This paper studied the geometric and combinatorial aspects of the classical Lawson's flip algorithm in 1972. Let A be a finite set of points in R2, omega be a height function which lifts the vertices of A into R3. Every flip in triangulations of A can be associated with a direction. We first established a relatively obvious relation between monotone sequences of directed flips between triangulations of A and triangulations of the lifted point set of A in R3. We then studied the structural properties of a directed flip graph (a poset) on the set of all triangulations of A. We proved several general properties of this poset which clearly explain when Lawson's algorithm works and why it may fail in general. We further characterised the triangulations which cause failure of Lawson's algorithm, and showed that they must contain redundant interior vertices which are not removable by directed flips. A special case if this result in 3d has been shown by B.Joe in 1989. As an application, we described a simple algorithm to triangulate a special class of 3d non-convex polyhedra. We proved sufficient conditions for the termination of this algorithm and show that it runs in O(n3) time.Comment: 40 pages, 35 figure

Bounds on the maximum multiplicity of some common geometric graphs

We obtain new lower and upper bounds for the maximum multiplicity of some weighted and, respectively, non-weighted common geometric graphs drawn on n points in the plane in general position (with no three points collinear): perfect matchings, spanning trees, spanning cycles (tours), and triangulations. (i) We present a new lower bound construction for the maximum number of triangulations a set of n points in general position can have. In particular, we show that a generalized double chain formed by two almost convex chains admits {\Omega}(8.65^n) different triangulations. This improves the bound {\Omega}(8.48^n) achieved by the double zig-zag chain configuration studied by Aichholzer et al. (ii) We present a new lower bound of {\Omega}(12.00^n) for the number of non-crossing spanning trees of the double chain composed of two convex chains. The previous bound, {\Omega}(10.42^n), stood unchanged for more than 10 years. (iii) Using a recent upper bound of 30^n for the number of triangulations, due to Sharir and Sheffer, we show that n points in the plane in general position admit at most O(68.62^n) non-crossing spanning cycles. (iv) We derive lower bounds for the number of maximum and minimum weighted geometric graphs (matchings, spanning trees, and tours). We show that the number of shortest non-crossing tours can be exponential in n. Likewise, we show that both the number of longest non-crossing tours and the number of longest non-crossing perfect matchings can be exponential in n. Moreover, we show that there are sets of n points in convex position with an exponential number of longest non-crossing spanning trees. For points in convex position we obtain tight bounds for the number of longest and shortest tours. We give a combinatorial characterization of the longest tours, which leads to an O(nlog n) time algorithm for computing them

Cambrian triangulations and their tropical realizations

This paper develops a Cambrian extension of the work of C. Ceballos, A. Padrol and C. Sarmiento on $\nu$-Tamari lattices and their tropical realizations. For any signature $\varepsilon \in \{\pm\}^n$, we consider a family of $\varepsilon$-trees in bijection with the triangulations of the $\varepsilon$-polygon. These $\varepsilon$-trees define a flag regular triangulation $\mathcal{T}^\varepsilon$ of the subpolytope $\operatorname{conv} \{(\mathbf{e}_{i_\bullet}, \mathbf{e}_{j_\circ}) \, | \, 0 \le i_\bullet < j_\circ \le n+1 \}$ of the product of simplices $\triangle_{\{0_\bullet, \dots, n_\bullet\}} \times \triangle_{\{1_\circ, \dots, (n+1)_\circ\}}$. The oriented dual graph of the triangulation $\mathcal{T}^\varepsilon$ is the Hasse diagram of the (type $A$) $\varepsilon$-Cambrian lattice of N. Reading. For any $I_\bullet \subseteq \{0_\bullet, \dots, n_\bullet\}$ and $J_\circ \subseteq \{1_\circ, \dots, (n+1)_\circ\}$, we consider the restriction $\mathcal{T}^\varepsilon_{I_\bullet, J_\circ}$ of the triangulation $\mathcal{T}^\varepsilon$ to the face $\triangle_{I_\bullet} \times \triangle_{J_\circ}$. Its dual graph is naturally interpreted as the increasing flip graph on certain $(\varepsilon, I_\bullet, J_\circ)$-trees, which is shown to be a lattice generalizing in particular the $\nu$-Tamari lattices in the Cambrian setting. Finally, we present an alternative geometric realization of $\mathcal{T}^\varepsilon_{I_\bullet, J_\circ}$ as a polyhedral complex induced by a tropical hyperplane arrangement.Comment: 16 pages, 11 figure

Geometric biplane graphs I: maximal graphs

We study biplane graphs drawn on a finite planar point set in general position. This is the family of geometric graphs whose vertex set is and can be decomposed into two plane graphs. We show that two maximal biplane graphs-in the sense that no edge can be added while staying biplane-may differ in the number of edges, and we provide an efficient algorithm for adding edges to a biplane graph to make it maximal. We also study extremal properties of maximal biplane graphs such as the maximum number of edges and the largest maximum connectivity over -element point sets.Peer ReviewedPostprint (author's final draft

Geometric biplane graphs I: maximal graphs

Postprint (author’s final draft

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

Geometric Biplane Graphs II: Graph Augmentation

We study biplane graphs drawn on a nite point set S in the plane in general position. This is the family of geometric graphs whose vertex set is S and which can be decomposed into two plane graphs. We show that every su ciently large point set admits a 5-connected biplane graph and that there are arbitrarily large point sets that do not admit any 6- connected biplane graph. Furthermore, we show that every plane graph (other than a wheel or a fan) can be augmented into a 4-connected biplane graph. However, there are arbitrarily large plane graphs that cannot be augmented to a 5-connected biplane graph by adding pairwise noncrossing edges.Peer ReviewedPostprint (author’s final draft