scholarly journals The Erdős-Sós conjecture for geometric graphs

2013 ◽  
Vol Vol. 15 no. 1 (Combinatorics) ◽  
Author(s):  
Luis Barba ◽  
Ruy Fabila-Monroy ◽  
Dolores Lara ◽  
Jesús Leaños ◽  
Cynthia Rodrıguez ◽  
...  
Keyword(s):  
Geometric Graph ◽  
Geometric Graphs ◽  
Convex Position ◽  
Minimum Number ◽  
International Audience ◽  
Set Of Points

Combinatorics International audience Let f(n,k) be the minimum number of edges that must be removed from some complete geometric graph G on n points, so that there exists a tree on k vertices that is no longer a planar subgraph of G. In this paper we show that ( 1 / 2 )n2 / k-1-n / 2≤f(n,k) ≤2 n(n-2) / k-2. For the case when k=n, we show that 2 ≤f(n,n) ≤3. For the case when k=n and G is a geometric graph on a set of points in convex position, we completely solve the problem and prove that at least three edges must be removed.

scholarly journals ON SPANNERS OF GEOMETRIC GRAPHS

2009 ◽  
Vol 20 (01) ◽  
pp. 135-149 ◽  
Author(s):  
JOACHIM GUDMUNDSSON ◽  
MICHIEL SMID
Keyword(s):  
Real Number ◽  
Upper Bound ◽  
Weighted Graph ◽  
Geometric Graph ◽  
Geometric Graphs ◽  
Np Hard ◽  
Minimum Number ◽  
Np Hardness

Given a connected geometric graph G, we consider the problem of constructing a t-spanner of G having the minimum number of edges. We prove that for every real number t with [Formula: see text], there exists a connected geometric graph G with n vertices, such that every t-spanner of G contains Ω(n1+1/t) edges. This bound almost matches the known upper bound, which states that every connected weighted graph with n vertices contains a t-spanner with O(n1+2/(t-1)) edges. We also prove that the problem of deciding whether a given geometric graph contains a t-spanner with at most K edges is NP-hard. Previously, this NP-hardness result was only known for non-geometric graphs.

scholarly journals Edge-Removal and Non-Crossing Configurations in Geometric Graphs

2010 ◽  
Vol Vol. 12 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Oswin Aichholzer ◽  
Sergio Cabello ◽  
Ruy Fabila-Monroy ◽  
David Flores-Peñaloza ◽  
Thomas Hackl ◽  
...  
Keyword(s):  
Extremal Problem ◽  
General Position ◽  
Geometric Graph ◽  
Perfect Matchings ◽  
Geometric Graphs ◽  
Line Segments ◽  
Straight Line ◽  
Edge Removal ◽  
International Audience ◽  
Point Set

Graphs and Algorithms International audience A geometric graph is a graph G = (V, E) drawn in the plane, such that V is a point set in general position and E is a set of straight-line segments whose endpoints belong to V. We study the following extremal problem for geometric graphs: How many arbitrary edges can be removed from a complete geometric graph with n vertices such that the remaining graph still contains a certain non-crossing subgraph. The non-crossing subgraphs that we consider are perfect matchings, subtrees of a given size, and triangulations. In each case, we obtain tight bounds on the maximum number of removable edges.

scholarly journals Cops and Robbers on Geometric Graphs

2012 ◽  
Vol 21 (6) ◽  
pp. 816-834 ◽  
Author(s):  
ANDREW BEVERIDGE ◽  
ANDRZEJ DUDEK ◽  
ALAN FRIEZE ◽  
TOBIAS MÜLLER
Keyword(s):  
Lower Bound ◽  
Absolute Constant ◽  
High Probability ◽  
Probability Space ◽  
Geometric Graph ◽  
Geometric Graphs ◽  
Random Geometric Graphs ◽  
Cops And Robbers ◽  
Minimum Number ◽  
Vertex Set

Cops and robbers is a turn-based pursuit game played on a graph G. One robber is pursued by a set of cops. In each round, these agents move between vertices along the edges of the graph. The cop number c(G) denotes the minimum number of cops required to catch the robber in finite time. We study the cop number of geometric graphs. For points x1, . . ., xn ∈ ℝ2, and r ∈ ℝ+, the vertex set of the geometric graph G(x1, . . ., xn; r) is the graph on these n points, with xi, xj adjacent when ∥xi − xj∥ ≤ r. We prove that c(G) ≤ 9 for any connected geometric graph G in ℝ2 and we give an example of a connected geometric graph with c(G) = 3. We improve on our upper bound for random geometric graphs that are sufficiently dense. Let (n,r) denote the probability space of geometric graphs with n vertices chosen uniformly and independently from [0,1]2. For G ∈ (n,r), we show that with high probability (w.h.p.), if r ≥ K1 (log n/n)1/4 then c(G) ≤ 2, and if r ≥ K2(log n/n)1/5 then c(G) = 1, where K1, K2 > 0 are absolute constants. Finally, we provide a lower bound near the connectivity regime of (n,r): if r ≤ K3 log n/ then c(G) > 1 w.h.p., where K3 > 0 is an absolute constant.

scholarly journals Packing 1-plane Hamiltonian cycles in complete geometric graphs

Filomat ◽  
2019 ◽  
Vol 33 (6) ◽  
pp. 1561-1574
Author(s):  
Hazim Trao ◽  
Niran Ali ◽  
Gek Chia ◽  
Adem Kilicman
Keyword(s):  
Lower Bound ◽  
General Position ◽  
Geometric Graph ◽  
Hamiltonian Cycles ◽  
Geometric Graphs ◽  
Convex Position

Counting the number of Hamiltonian cycles that are contained in a geometric graph is #P-complete even if the graph is known to be planar. A relaxation for problems in plane geometric graphs is to allow the geometric graphs to be 1-plane, that is, each of its edges is crossed at most once. We consider the following question: For any set P of n points in the plane, how many 1-plane Hamiltonian cycles can be packed into a complete geometric graph Kn? We investigate the problem by taking three different situations of P, namely, when P is in convex position and when P is in wheel configurations position. Finally, for points in general position we prove the lower bound of k - 1 where n = 2k + h and 0 ? h < 2k. In all of the situations, we investigate the constructions of the graphs obtained.

scholarly journals Colouring random geometric graphs

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Colin J. H. McDiarmid ◽  
Tobias Müller
Keyword(s):  
Chromatic Number ◽  
Analogous Result ◽  
Clique Number ◽  
Geometric Graph ◽  
Geometric Graphs ◽  
Random Geometric Graph ◽  
Random Geometric Graphs ◽  
Different Types ◽  
International Audience ◽  
Uniform Case

International audience A random geometric graph $G_n$ is obtained as follows. We take $X_1, X_2, \ldots, X_n ∈\mathbb{R}^d$ at random (i.i.d. according to some probability distribution ν on $\mathbb{R}^d$). For $i ≠j$ we join $X_i$ and $X_j$ by an edge if $║X_i - X_j ║< r(n)$. We study the properties of the chromatic number $χ _n$ and clique number $ω _n$ of this graph as n becomes large, where we assume that $r(n) →0$. We allow any choice $ν$ that has a bounded density function and $║. ║$ may be any norm on $ℝ^d$. Depending on the choice of $r(n)$, qualitatively different types of behaviour can be observed. We distinguish three main cases, in terms of the key quantity $n r^d$ (which is a measure of the average degree). If $r(n)$ is such that $\frac{nr^d}{ln n} →0$ as $n →∞$ then $\frac{χ _n}{ ω _n} →1$ almost surely. If n $\frac{r^d }{\ln n} →∞$ then $\frac{χ _n }{ ω _n} →1 / δ$ almost surely, where $δ$ is the (translational) packing density of the unit ball $B := \{ x ∈ℝ^d: ║x║< 1 \}$ (i.e. $δ$ is the proportion of $d$-space that can be filled with disjoint translates of $B$). If $\frac{n r^d }{\ln n} →t ∈(0,∞)$ then $\frac{χ _n }{ ω _n}$ tends almost surely to a constant that can be bounded in terms of $δ$ and $t$. These results extend earlier work of McDiarmid and Penrose. The proofs in fact yield separate expressions for $χ _n$ and $ω _n$. We are also able to prove a conjecture by Penrose. This states that when $\frac{n r^d }{ln n} →0$ then the clique number becomes focussed on two adjacent integers, meaning that there exists a sequence $k(n)$ such that $\mathbb{P}( ω _n ∈\{k(n), k(n)+1\}) →1$ as $n →∞$. The analogous result holds for the chromatic number (and for the maximum degree, as was already shown by Penrose in the uniform case).

scholarly journals Ordinary hyperspheres and spherical curves

2021 ◽  
Vol 21 (1) ◽  
pp. 15-22
Author(s):  
Aaron Lin ◽  
Konrad Swanepoel
Keyword(s):  
Degenerate Case ◽  
Minimum Number ◽  
Set Of Points ◽  
Orchard Problem

Abstract An ordinary hypersphere of a set of points in real d-space, where no d + 1 points lie on a (d - 2)-sphere or a (d - 2)-flat, is a hypersphere (including the degenerate case of a hyperplane) that contains exactly d + 1 points of the set. Similarly, a (d + 2)-point hypersphere of such a set is one that contains exactly d + 2 points of the set. We find the minimum number of ordinary hyperspheres, solving the d-dimensional spherical analogue of the Dirac–Motzkin conjecture for d ⩾ 3. We also find the maximum number of (d + 2)-point hyperspheres in even dimensions, solving the d-dimensional spherical analogue of the orchard problem for even d ⩾ 4.

scholarly journals Clique Colourings of Geometric Graphs

10.37236/7159 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Colin McDiarmid ◽  
Dieter Mitsche ◽  
Pawel Prałat
Keyword(s):  
Asymptotic Behaviour ◽  
Chromatic Number ◽  
Euclidean Distance ◽  
High Probability ◽  
Maximal Clique ◽  
Higher Dimensions ◽  
Geometric Graph ◽  
Geometric Graphs ◽  
Random Geometric Graph ◽  
Vertex Set

A clique colouring of a graph is a colouring of the vertices such that no maximal clique is monochromatic (ignoring isolated vertices). The least number of colours in such a colouring is the clique chromatic number.  Given $n$ points $\mathbf{x}_1, \ldots,\mathbf{x}_n$ in the plane, and a threshold $r>0$, the corresponding geometric graph has vertex set $\{v_1,\ldots,v_n\}$, and distinct $v_i$ and $v_j$ are adjacent when the Euclidean distance between $\mathbf{x}_i$ and $\mathbf{x}_j$ is at most $r$. We investigate the clique chromatic number of such graphs.We first show that the clique chromatic number is at most 9 for any geometric graph in the plane, and briefly consider geometric graphs in higher dimensions. Then we study the asymptotic behaviour of the clique chromatic number for the random geometric graph $\mathcal{G}$ in the plane, where $n$ random points are independently and uniformly distributed in a suitable square. We see that as $r$ increases from 0, with high probability the clique chromatic number is 1 for very small $r$, then 2 for small $r$, then at least 3 for larger $r$, and finally drops back to 2.

scholarly journals Strong chromatic index of products of graphs

2007 ◽  
Vol Vol. 9 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Olivier Togni
Keyword(s):  
Cartesian Product ◽  
Complete Graphs ◽  
Chromatic Index ◽  
Induced Matching ◽  
Colour Class ◽  
Strong Chromatic Index ◽  
Minimum Number ◽  
International Audience

Graphs and Algorithms International audience The strong chromatic index of a graph is the minimum number of colours needed to colour the edges in such a way that each colour class is an induced matching. In this paper, we present bounds for strong chromatic index of three different products of graphs in term of the strong chromatic index of each factor. For the cartesian product of paths, cycles or complete graphs, we derive sharper results. In particular, strong chromatic indices of d-dimensional grids and of some toroidal grids are given along with approximate results on the strong chromatic index of generalized hypercubes.

APPROXIMATION ALGORITHMS FOR A VARIANT OF DISCRETE PIERCING SET PROBLEM FOR UNIT DISKS

2013 ◽  
Vol 23 (06) ◽  
pp. 461-477 ◽  
Author(s):  
MINATI DE ◽  
GAUTAM K. DAS ◽  
PAZ CARMI ◽  
SUBHAS C. NANDY
Keyword(s):  
Approximation Algorithms ◽  
Simple Algorithm ◽  
Constant Factor ◽  
Performance Ratio ◽  
Approximation Result ◽  
Worst Case ◽  
Approximation Factor ◽  
Minimum Number ◽  
Unit Disks ◽  
Set Of Points

In this paper, we consider constant factor approximation algorithms for a variant of the discrete piercing set problem for unit disks. Here a set of points P is given; the objective is to choose minimum number of points in P to pierce the unit disks centered at all the points in P. We first propose a very simple algorithm that produces 12-approximation result in O(n log n) time. Next, we improve the approximation factor to 4 and then to 3. The worst case running time of these algorithms are O(n8 log n) and O(n15 log n) respectively. Apart from the space required for storing the input, the extra work-space requirement for each of these algorithms is O(1). Finally, we propose a PTAS for the same problem. Given a positive integer k, it can produce a solution with performance ratio [Formula: see text] in nO(k) time.

scholarly journals On Geometric Graphs with No Two Edges in Convex Position

1998 ◽  
Vol 19 (3) ◽  
pp. 399-404 ◽  
Author(s):  
M. Katchalski ◽  
H. Last
Keyword(s):  
Geometric Graphs ◽  
Convex Position
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