Therefore, we look at orientations with prescribed degrees, bipolar orientations, separating decompositions, and transversal structures. In the second part, we ask for representations by a dissections of a rectangle into squares.
We review results by Brooks et al. Isr J Math —84 ], and Schramm [Square tilings with prescribed combinatorics. Isr J Math — ] and discuss a technique of computing squarings via solutions of systems of linear equations. We show that the disjoint convex obstacle number of an outerplanar graph is always at most 5, and of a bipartite permutation graph at most 4. The former answers a question raised by Alpert, Koch, and Laison. We complement the upper bound for outerplanar graphs with the lower bound of 4. We give a new proof of this result and strengthen it by showing that the conclusion remains true even if adjacent edges are allowed to cross each other oddly.
We obtain a new and simple algorithm to test level-planarity in quadratic time, and we show that. A geometric graph contains a. We also settle the conjecture for geometric graphs with no 2, 1 -crossing family. As a direct application, this implies that for any circle graph. We generalize the notions of flippable and simultaneously flippable edges in a triangulation of a set.
We prove a worst-case tight lower bound for the number of pseudo-simultaneously flippable edges in a triangulation in terms of the number of vertices. We use this bound for deriving new upper bounds for the maximal number of crossing-free straight-edge graphs that can be embedded on any fixed set of. We obtain new upper bounds for the number of spanning trees and forests as well. Specifically, let. We also obtain upper bounds for the number of crossing-free straight-edge graphs that have.
Graph augmentation problems are motivated by network design and have been studied extensively in optimization. We consider augmentation problems over plane geometric graphs, that is, graphs given with a crossing-free straight-line embedding in the plane. The geometric constraints on the possible new edges render some of the simplest augmentation problems intractable, and in many cases only extremal results are known. We survey recent results, highlight common trends, and gather numerous conjectures and open problems. There are some results on geometric graphs on a set of red and blue points in the plane.
We show that some similar results also hold for a set of red and blue points in the plane lattice using. We survey some results and collect a set of open problems related to graph Ramsey theory with geometric constraints. In this chapter, we present a complete characterization of the smallest sets that block all the simple spanning trees SSTs in a complete geometric graph.
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We also show that if a subgraph is a blocker for all SSTs of diameter at most 4, then it must block all simple spanning subgraphs and, in particular, all SSTs. For convex geometric graphs, we obtain an even stronger result: Being a blocker for all SSTs of diameter at most 3 is already sufficient for blocking all simple spanning subgraphs.
The best-known general upper bound on the chromatic number of circle graphs with clique number. Based on this result, we prove that the chromatic number of every circle graph with clique number at most 3 is at most Our main approach revolves around a few known facts about distances, together with a computer program that searches all distance configurations of two disjoint convex hull intervals up to some finite size. We thereby obtain other new bounds, such as.
In this chapter, we discuss two classical problems lying on the edge of graph theory and combinatorial geometry. The first problem is due to E. It consists of coloring metric spaces in such a way that pairs of points at some prescribed distances receive different colors. The second problem is attributed to K. Borsuk and involves finding the minimum number of parts of smaller diameter into which an arbitrary bounded nonsingleton point set in a metric space can be partitioned.
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Both problems are easily translated into the language of graph theory, provided we consider, instead of the whole space, any finite distance graph or any finite graph of diameters. During the last decades, a huge number of ideas have been proposed for solving both problems, and many results in both directions have been obtained.
In the survey below, we try to give an entire picture of this beautiful area of geometric combinatorics. We show that deciding whether a graph with given edge lengths can be realized by a straight-line drawing has the same complexity as deciding the truth of sentences in the existential theory of the real numbers,.
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Linkage realizability is. Testing whether a configuration is not rigid is. The two main theorems are 1 every. These results support two conjectures that were raised by Dey and Pach, and by Akiyama and Alon.
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The implied constants are absolute. The task of this survey is to present various results on intersection patterns of convex sets. One of the main tools for studying intersection patterns is a point of view via simplicial complexes. We recall the definitions of so-called. We study the differences among these notions and also focus on computational complexity for recognizing them.
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A list of Helly-type theorems is presented in the survey. We also discuss the important role played by the above-mentioned notions for the theorems. We mainly focus on new results. A graph drawn in the plane with straight-line edges is called a geometric graph. If no path of length at most.
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For the proof, we develop randomized thinning procedures for edge-colored bipartite abstract graphs that can be applied to other problems as well. Springer Professional. Back to the search result list. Table of Contents Frontmatter.
In the present volume, by a geometric graph we mean a graph drawn in the plane so that its vertices are represented by distinct points and its edges by possibly crossing straight-line segments between these points such that no edge passes through a vertex different from its endpoints. Topological graphs are defined analogously, except that their edges can be represented by simple Jordan arcs .
This paper studies problems related to visibility among points in the plane. The focus of this paper is the conjecture that each k -blocked set has bounded size as a function of k. Among other results, for infinitely many values of k , we construct k -blocked sets with k 1. We also examine possible conditions under which this bound may be improved, such as when G 0 is a collection of interior-disjoint convex polygons in a triangular container. Souvaine, Csaba D.
Let P be a set of n points in the plane. A topological hypergraphG , on the set of points of P , is a collection of simple closed curves in the plane that avoid the points of P. Each of these curves is called an edge of G , and the points of P are called the vertices of G. Let P be a set of points in the plane in general position. How many triangulations of P are needed to cover all the empty triangles of P?
We also study the following problem: What is the largest number of edge-disjoint triangles of P containing a point q of the plane in their interior? Learn More - opens in a new window or tab International shipping and import charges paid to Pitney Bowes Inc.
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