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### Note on Geometric Graphs

Géza Tóth
2000 Journal of combinatorial theory. Series A
We show that a geometric graph of n vertices with no k + 1 pairwise disjoint edges has at most 2 9 k 2 n edges.  ...  A geometric graph is a graph drawn in the plane so that the vertices are represented by points in general position, the edges are represented by straight line segments connecting the corresponding points  ...  In this note we further improve the upper bound. Theorem 1. For any k < n/2, e k (n) ≤ 2 9 k 2 n. Let G be a geometric graph.  ...

### A note on light geometric graphs

Eyal Ackerman, Jacob Fox, Rom Pinchasi
2013 Discrete Mathematics
Let G be a geometric graph on n vertices in general position in the plane.  ...  We extend the previous result in  and with a shorter argument show that every k-light geometric graph on n vertices has at most O(n √ k) edges. This bound is best possible.  ...  Lemma 2. 1 . 1 Let G be an oriented geometric graph on n vertices.  ...

### A note on harmonic subgraphs in labelled geometric graphs

G. Araujo, J. Balogh, R. Fabila, G. Salazar, J. Urrutia
2008 Information Processing Letters
In this note we investigate the maximum size of noncrossing matchings and paths on S, under the requirement that no two edges have the same weight.  ...  Introduction The study of geometric graphs on point sets in the plane has received a considerable amount of attention lately [6, 8] .  ...  Note that in these questions, the labelling map is two-to-one.  ...

### On Spanners of Geometric Graphs [chapter]

Joachim Gudmundsson, Michiel Smid
2006 Lecture Notes in Computer Science
Previously, this NP-hardness result was only known for non-geometric graphs.  ...  Given a connected geometric graph G, we consider the problem of constructing a tspanner of G having the minimum number of edges.  ...  Workshop on Algorithm Theory, Lecture Notes in Computer Science, Volume 4059, Springer-Verlag, 2006, 388-399.  ...

### Parameterized Complexity of Independence and Domination on Geometric Graphs [chapter]

Dániel Marx
2006 Lecture Notes in Computer Science
We investigate the parameterized complexity of Maximum Independent Set and Dominating Set restricted to certain geometric graphs.  ...  We show that Dominating Set is W-hard for the intersection graphs of unit squares, unit disks, and line segments.  ...  Therefore, it is an interesting question to investigate the complexity of these problems on different types of geometric graphs.  ...

### On the Cover Time of Random Geometric Graphs [chapter]

Chen Avin, Gunes Ercal
2005 Lecture Notes in Computer Science
A random geometric graph G(n, r) is obtained by placing n points uniformly at random on the unit square and connecting two points iff their Euclidean distance is at most r.  ...  Recently, with the advent of ad-hoc and sensor networks, an interesting class of random graphs, namely random geometric graphs, has gained new relevance and its properties have been the subject of much  ...  Note immediately last corollary that this yields negative results for G 1 (n, r) in that optimality of cover time for one-dimensional geometric graphs requires a radius of order strictly greater than the  ...

### On Paths in a Complete Bipartite Geometric Graph [chapter]

Atsushi Kaneko, M. Kano
2001 Lecture Notes in Computer Science
A geometric complete bipartite graph K(A, B) is a complete bipartite graph with partite sets A and B which is drawn in the plane such that each edge of K (A, B) is a straight-line segment.  ...  We prove that (i) If |B| ≥ (n + 1)(2n − 4) + 1, then the geometric complete bipartite graph K(A, B) contains a path that passes through all the points in A and has no crossings; and (ii) There exists a  ...  If a geometric graph G is a complete bipartite graph with partite sets A and B, i.e., V (G) = A ∪ B, then G is denoted by K(A, B), which may be called a geometric complete bipartite graph.  ...

### Large Scale Simulations of a Neural Network Model for the Graph Bisection Problem on Geometrically Connected Graphs

Gonzalo Hernandez, Luis Salinas
2004 Electronic Notes in Discrete Mathematics
In this work some preliminary numerical results obtained by large scale simulations of the sequential dynamics of a neural network model for the graph bisection problem on random geometrically connected  ...  graphs are presented.  ...  geometrically connected graphs.  ...

### Small Worlds and Rapid Mixing with a Little More Randomness on Random Geometric Graphs [chapter]

Gunes Ercal
2011 Lecture Notes in Computer Science
Let the G1 new wiring be such that we form a random cubic graph amongst the bin-leaders and superimpose this upon the random geometric graph.  ...  Let the G2 new wiring be such that we form a random 1-out graph amongst the bin-leaders and superimpose this upon the random geometric graph.  ...  Either random or deterministic. 2 Note that if a geometric graph is μ1-geo-dense then it is also μ2-geo-dense for any μ2 < μ1.  ...

### On Geometric Structure of Global Roundings for Graphs and Range Spaces [chapter]

Tetsuo Asano, Naoki Katoh, Hisao Tamaki, Takeshi Tokuyama
2004 Lecture Notes in Computer Science
path hypergraph of a series-parallel graph.  ...  We study geometric (or combinatorial) structure of the set of global roundings of a using the notion of compatible set with respect to the discrepancy distance.  ...  Also, they thank Ken-ichi Kawarabayashi and Takao Nishizeki for providing knowledge on the structure of series-parallel graphs.  ...

### A note on a theorem of Perles concerning non-crossing paths in convex geometric graphs

Ana Paulina Figueroa
2009 Computational geometry
Acknowledgements I would like to thank Eduardo Rivera-Campo, Gyula Károlyi and an anonymous referee for their suggestions about the presentation of this note.  ...  Note that |Z | = 2|P |.  ...  If P is in convex position then G is called a convex geometric graph. A non-crossing path in a geometric graph is a path which does not contain any intersecting pair of edges.  ...

### On Local Transformations in Plane Geometric Graphs Embedded on Small Grids [chapter]

Manuel Abellanas, Prosenjit Bose, Alfredo García, Ferran Hurtado, Pedro Ramos, Eduardo Rivera-Campo, Javier Tejel
2004 Lecture Notes in Computer Science
Given two n-vertex plane graphs G 1 = (V 1 , E 1 ) and G 2 = (V 2 , E 2 ) with |E 1 | = |E 2 | embedded in the n × n grid, with straightline segments as edges, we show that with a sequence of O(n) point  ...  Note that there is a discrepancy between the combinatorial result and the geometric one.  ...  Note that point moves do not change the connectivity of a graph. Therefore, we solely need to concentrate on edge moves. The main tool we used for edge moves in triangulations is Lemma 1.  ...

### Bounds on the Geometric Mean of Arc Lengths for Bounded-Degree Planar Graphs [chapter]

Mohammad Khairul Hasan, Sung-Eui Yoon, Kyung-Yong Chwa
2009 Lecture Notes in Computer Science
Cache-oblivious layouts, constructed to minimize the geometric mean of arc lengths of graphs, have been adapted to reduce data access time during random walks on graphs.  ...  In this paper, we present a constant factor approximation algorithm for the Minimum Geometric Mean Layout (MGML) problem for bounded-degree planar graphs.  ...  This observation strongly supported that there is possibility that we can have a constant-factor approximation algorithm in terms of minimizing the geometric means.  ...

### The Domination Number of On-line Social Networks and Random Geometric Graphs [chapter]

Anthony Bonato, Marc Lozier, Dieter Mitsche, Xavier Pérez-Giménez, Paweł Prałat
2015 Lecture Notes in Computer Science
Asymptotic sublinear bounds are rigorously derived for the domination number of graphs generated by the memoryless geometric protean random graph model.  ...  In addition, we derive the asymptotic value of the domination number in classical random geometric graphs.  ...  The basic reference on random geometric graphs is the monograph by Penrose  .  ...

### On Radio Broadcasting in Random Geometric Graphs [chapter]

Robert Elsässer, Leszek Gąsieniec, Thomas Sauerwald
Lecture Notes in Computer Science
Since this graph is usually not strongly connected, we assume that the message which has to be spread to all nodes of the graph is placed initially in one of the nodes of the giant component.  ...  In this paper we consider radio broadcasting in random geometric graphs, in which n nodes are placed uniformly at random in [0, √ n] 2 , and there is a (directed) edge from a node u to a node v in the  ...  The results of Theorems 3 and 4 can be extended to further random geometric graph models.  ...
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