Maximizing Maximal Angles for Plane Straight-Line Graphs

Let $G=(S, E)$ be a plane straight-line graph on a finite point set $S\subset\R^2$ in general position. The incident angles of a vertex $p \in S$ of $G$ are the angles between any two edges of $G$ that appear consecutively in the circular order of the edges incident to $p$. A plane straight-line graph is called $\phi$-open if each vertex has an incident angle of size at least $\phi$. In this paper we study the following type of question: What is the maximum angle $\phi$ such that for any finite set $S\subset\R^2$ of points in general position we can find a graph from a certain class of graphs on $S$ that is $\phi$-open? In particular, we consider the classes of triangulations, spanning trees, and paths on $S$ and give tight bounds in most cases.Comment: 15 pages, 14 figures. Apart of minor corrections, some proofs that were omitted in the previous version are now include

Flip Graphs of Degree-Bounded (Pseudo-)Triangulations

We study flip graphs of triangulations whose maximum vertex degree is bounded by a constant $k$. In particular, we consider triangulations of sets of $n$ points in convex position in the plane and prove that their flip graph is connected if and only if $k > 6$; the diameter of the flip graph is $O(n^2)$. We also show that, for general point sets, flip graphs of pointed pseudo-triangulations can be disconnected for $k \leq 9$, and flip graphs of triangulations can be disconnected for any $k$. Additionally, we consider a relaxed version of the original problem. We allow the violation of the degree bound $k$ by a small constant. Any two triangulations with maximum degree at most $k$ of a convex point set are connected in the flip graph by a path of length $O(n \log n)$, where every intermediate triangulation has maximum degree at most $k+4$.Comment: 13 pages, 12 figures, acknowledgments update

Computational Geometry Column 43

The concept of pointed pseudo-triangulations is defined and a few of its applications described.Comment: 3 pages, 1 figur

Tight triangulations of closed 3-manifolds

It is well known that a triangulation of a closed 2-manifold is tight with respect to a field of characteristic two if and only if it is neighbourly; and it is tight with respect to a field of odd characteristic if and only if it is neighbourly and orientable. No such characterization of tightness was previously known for higher dimensional manifolds. In this paper, we prove that a triangulation of a closed 3-manifold is tight with respect to a field of odd characteristic if and only if it is neighbourly, orientable and stacked. In consequence, the K\"{u}hnel-Lutz conjecture is valid in dimension three for fields of odd characteristic. Next let $\mathbb{F}$ be a field of characteristic two. It is known that, in this case, any neighbourly and stacked triangulation of a closed 3-manifold is $\mathbb{F}$-tight. For triangulated closed 3-manifolds with at most 71 vertices or with first Betti number at most 188, we show that the converse is true. But the possibility of an $\mathbb{F}$-tight non-stacked triangulation on a larger number of vertices remains open. We prove the following upper bound theorem on such triangulations. If an $\mathbb{F}$-tight triangulation of a closed 3-manifold has $n$ vertices and first Betti number $\beta_1$, then $(n-4)(617n- 3861) \leq 15444\beta_1$. Equality holds here if and only if all the vertex links of the triangulation are connected sums of boundary complexes of icosahedra.Comment: 21 pages, 1 figur

An Improved Lower Bound on the Minimum Number of Triangulations

Upper and lower bounds for the number of geometric graphs of specific types on a given set of points in the plane have been intensively studied in recent years. For most classes of geometric graphs it is now known that point sets in convex position minimize their number. However, it is still unclear which point sets minimize the number of geometric triangulations; the so-called double circles are conjectured to be the minimizing sets. In this paper we prove that any set of n points in general position in the plane has at least Omega(2.631^n) geometric triangulations. Our result improves the previously best general lower bound of Omega(2.43^n) and also covers the previously best lower bound of Omega(2.63^n) for a fixed number of extreme points. We achieve our bound by showing and combining several new results, which are of independent interest: (1) Adding a point on the second convex layer of a given point set (of 7 or more points) at least doubles the number of triangulations. (2) Generalized configurations of points that minimize the number of triangulations have at most n/2 points on their convex hull. (3) We provide tight lower bounds for the number of triangulations of point sets with up to 15 points. These bounds further support the double circle conjecture

One brick at a time: a survey of inductive constructions in rigidity theory

We present a survey of results concerning the use of inductive constructions to study the rigidity of frameworks. By inductive constructions we mean simple graph moves which can be shown to preserve the rigidity of the corresponding framework. We describe a number of cases in which characterisations of rigidity were proved by inductive constructions. That is, by identifying recursive operations that preserved rigidity and proving that these operations were sufficient to generate all such frameworks. We also outline the use of inductive constructions in some recent areas of particularly active interest, namely symmetric and periodic frameworks, frameworks on surfaces, and body-bar frameworks. We summarize the key outstanding open problems related to inductions.Comment: 24 pages, 12 figures, final versio