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Plane geometric graph augmentation: a generic perspective
, 2011
"... 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 crossingfree straightline embedding in the plane. The geometric constraints on the possible ne ..."
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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 crossingfree straightline 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.
Maximizing Maximal Angles for Plane Straight Line Graphs
"... Let G =(S, E) be a plane straight line graph on a finite point set S ⊂ R 2 in general position. For a point p ∈ S let the maximum incident angle of p in G be the maximum angle between any two edges of G that appear consecutively in the circular order of the edges incident to p. A plane straight line ..."
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Let G =(S, E) be a plane straight line graph on a finite point set S ⊂ R 2 in general position. For a point p ∈ S let the maximum incident angle of p in G be the maximum angle 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 ϕopen if each vertex has an incident angle of size at least ϕ. In this paper we study the following type of question: What is the maximum angle ϕ such that for any finite set S ⊂ R 2 of points in general position we can find a graph from a certain class of graphs on S that is ϕopen? In particular, we consider the classes of triangulations, spanning trees, and paths on S and give tight bounds in most cases.
Realization of simply connected polygonal linkages and recognition of unit disk contact trees
"... Abstract. We wish to decide whether a simply connected flexible polygonal structure can be realized in Euclidean space. Two models are considered: polygonal linkages (bodyandjoint framework) and contact graphs of unit disks in the plane. (1) We show that it is strongly NPhard to decide whether a ..."
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Abstract. We wish to decide whether a simply connected flexible polygonal structure can be realized in Euclidean space. Two models are considered: polygonal linkages (bodyandjoint framework) and contact graphs of unit disks in the plane. (1) We show that it is strongly NPhard to decide whether a given polygonal linkage is realizable in the plane when the bodies are convex polygons and their contact graph is a tree; the problem is weakly NPhard already for a chain of rectangles, but efficiently decidable for a chain of triangles hinged at distinct vertices. (2) We also show that it is strongly NPhard to decide whether a given tree is the contact graph of interiordisjoint unit disks in the plane. 1
On Numbers of PseudoTriangulations∗
, 2014
"... We study the maximum numbers of pseudotriangulations and pointed pseudotriangulations that can be embedded over a specific set of points in the plane or contained in a specific triangulation. We derive the bounds O(5.45N) and Ω(2.41N) for the maximum number of pointed pseudotriangulations that c ..."
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We study the maximum numbers of pseudotriangulations and pointed pseudotriangulations that can be embedded over a specific set of points in the plane or contained in a specific triangulation. We derive the bounds O(5.45N) and Ω(2.41N) for the maximum number of pointed pseudotriangulations that can be contained in a specific triangulation over a set of N points. For the number of all pseudotriangulations contained in a triangulation we derive the bounds O∗(6.54N) and Ω(3.30N). We also prove that O∗(89.1N) pointed pseudotriangulations can be embedded over any specific set of N points in the plane, and at most 120N general pseudotriangulations. 1