On the number of spanning trees a planar graph can have
Kevin Buchin,André Schulz +1 more
- pp 110-121
Reads0
Chats0
TLDR
It is proved that any planar graph on n vertices has less than O(5.2852n) spanning trees and the grid size needed to realize a 3d polytope with integer coordinates can be bounded by O(147.7n).Abstract:
We prove that any planar graph on n vertices has less than O(5.2852n) spanning trees. Under the restriction that the planar graph is 3-connected and contains no triangle and no quadrilateral the number of its spanning trees is less than O(2.7156n). As a consequence of the latter the grid size needed to realize a 3d polytope with integer coordinates can be bounded by O(147.7n). Our observations imply improved upper bounds for related quantities: the number of cycle-free graphs in a planar graph is bounded by O(6.4884n), the number of plane spanning trees on a set of n points in the plane is bounded by O(158.6n), and the number of plane cycle-free graphs on a set of n points in the plane is bounded by O(194.7n).read more
Citations
More filters
Journal ArticleDOI
Counting Triangulations of Planar Point Sets
Micha Sharir,Adam Sheffer +1 more
TL;DR: The maximal number of triangulations that a planar set of $n$ points can have is shown to be at most $30^n, which can be used to derive new upper bounds for the number of planar graphs, spanning cycles, spanning trees, and cycle-free graphs.
Journal ArticleDOI
Tree Topology Estimation
TL;DR: A heuristic search algorithm is presented to estimate the most likely topology of a rooted, three-dimensional tree from a single two-dimensional image using a generative, parametric tree-growth model.
Journal ArticleDOI
Bounds on the Maximum Multiplicity of Some Common Geometric Graphs
TL;DR: New lower and upper bounds for the maximum multiplicity of some weighted and, respectively, nonweighted 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 are obtained.
Posted Content
Counting Triangulations of Planar Point Sets
Micha Sharir,Adam Sheffer +1 more
TL;DR: In this paper, the maximal number of triangulations that a planar set of points can have, and show that it is at most $30^n, was shown by a careful optimization of the charging scheme of Sharir and Welzl (2006), which led to the previous best upper bound of $43^n$ for the problem.
Posted Content
Counting Plane Graphs: Flippability and its Applications
TL;DR: In this paper, a worst-case lower bound for the number of pseudo-simultaneously flippable edges in a triangulation of a set S of points in the plane was shown.
References
More filters
Book
Matrix Analysis
Roger A. Horn,Charles R. Johnson +1 more
TL;DR: In this article, the authors present results of both classic and recent matrix analyses using canonical forms as a unifying theme, and demonstrate their importance in a variety of applications, such as linear algebra and matrix theory.
Journal ArticleDOI
Introduction to algorithms: 4. Turtle graphics
TL;DR: In this article, a language similar to logo is used to draw geometric pictures using this language and programs are developed to draw geometrical pictures using it, which is similar to the one we use in this paper.