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Register a new userGeometric Biplane Graphs I: Maximal Graphs
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Added by
Matias Korman
Publication date
2017
fields
Informatics Engineering
and research's language is
English
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We study biplane graphs drawn on a finite planar point set $S$ in general position. This is the family of geometric graphs whose vertex set is $S$ and can be decomposed into two plane graphs. We show that two maximal biplane graphs---in the sense that no edge can be added while staying biplane---may differ in the number of edges, and we provide an efficient algorithm for adding edges to a biplane graph to make it maximal. We also study extremal properties of maximal biplane graphs such as the maximum number of edges and the largest maximum connectivity over $n$-element point sets.
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We study biplane graphs drawn on a finite point set $S$ in the plane in general position. This is the family of geometric graphs whose vertex set is $S$ and which can be decomposed into two plane graphs. We show that every sufficiently large point set admits a 5-connected biplane graph and that there are arbitrarily large point sets that do not admit any 6-connected biplane graph. Furthermore, we show that every plane graph (other than a wheel or a fan) can be augmented into a 4-connected biplane graph. However, there are arbitrarily large plane graphs that cannot be augmented to a 5-connected biplane graph by adding pairwise noncrossing edges.
Efficient algorithms are presented for constructing spanners in geometric intersection graphs. For a unit ball graph in R^k, a (1+epsilon)-spanner is obtained using efficient partitioning of the space into hypercubes and solving bichromatic closest pair problems. The spanner construction has almost equivalent complexity to the construction of Euclidean minimum spanning trees. The results are extended to arbitrary ball graphs with a sub-quadratic running time. For unit ball graphs, the spanners have a small separator decomposition which can be used to obtain efficient algorithms for approximating proximity problems like diameter and distance queries. The results on compressed quadtrees, geometric graph separators, and diameter approximation might be of independent interest.
Let $mathcal{D}$ be a set of $n$ disks in the plane. The disk graph $G_mathcal{D}$ for $mathcal{D}$ is the undirected graph with vertex set $mathcal{D}$ in which two disks are joined by an edge if and only if they intersect. The directed transmission graph $G^{rightarrow}_mathcal{D}$ for $mathcal{D}$ is the directed graph with vertex set $mathcal{D}$ in which there is an edge from a disk $D_1 in mathcal{D}$ to a disk $D_2 in mathcal{D}$ if and only if $D_1$ contains the center of $D_2$. Given $mathcal{D}$ and two non-intersecting disks $s, t in mathcal{D}$, we show that a minimum $s$-$t$ vertex cut in $G_mathcal{D}$ or in $G^{rightarrow}_mathcal{D}$ can be found in $O(n^{3/2}text{polylog} n)$ expected time. To obtain our result, we combine an algorithm for the maximum flow problem in general graphs with dynamic geometric data structures to manipulate the disks. As an application, we consider the barrier resilience problem in a rectangular domain. In this problem, we have a vertical strip $S$ bounded by two vertical lines, $L_ell$ and $L_r$, and a collection $mathcal{D}$ of disks. Let $a$ be a point in $S$ above all disks of $mathcal{D}$, and let $b$ a point in $S$ below all disks of $mathcal{D}$. The task is to find a curve from $a$ to $b$ that lies in $S$ and that intersects as few disks of $mathcal{D}$ as possible. Using our improved algorithm for minimum cuts in disk graphs, we can solve the barrier resilience problem in $O(n^{3/2}text{polylog} n)$ expected time.
Online routing in a planar embedded graph is central to a number of fields and has been studied extensively in the literature. For most planar graphs no $O(1)$-competitive online routing algorithm exists. A notable exception is the Delaunay triangulation for which Bose and Morin [Online routing in triangulations. SIAM Journal on Computing, 33(4):937-951, 2004] showed that there exists an online routing algorithm that is $O(1)$-competitive. However, a Delaunay triangulation can have $Omega(n)$ vertex degree and a total weight that is a linear factor greater than the weight of a minimum spanning tree. We show a simple construction, given a set $V$ of $n$ points in the Euclidean plane, of a planar geometric graph on $V$ that has small weight (within a constant factor of the weight of a minimum spanning tree on $V$), constant degree, and that admits a local routing strategy that is $O(1)$-competitive. Moreover, the technique used to bound the weight works generally for any planar geometric graph whilst preserving the admission of an $O(1)$-competitive routing strategy.
Random graphs are mathematical models that have applications in a wide range of domains. We study the following model where one adds ErdH{o}s--Renyi (ER) type perturbation to a random geometric graph. More precisely, assume $G_mathcal{X}^{*}$ is a random geometric graph sampled from a nice measure on a metric space $mathcal{X} = (X,d)$. The input observed graph $widehat{G}(p,q)$ is generated by removing each existing edge from $G_mathcal{X}^*$ with probability $p$, while inserting each non-existent edge to $G_mathcal{X}^{*}$ with probability $q$. We refer to such random $p$-deletion and $q$-insertion as ER-perturbation. Although these graphs are related to the objects in the continuum percolation theory, our understanding of them is still rather limited. In this paper we consider a localized version of the classical notion of clique number for the aforementioned ER-perturbed random geometric graphs: Specifically, we study the edge clique number for each edge in a graph, defined as the size of the largest clique(s) in the graph containing that edge. The clique number of the graph is simply the largest edge clique number. Interestingly, given a ER-perturbed random geometric graph, we show that the edge clique number presents two fundamentally different types of behaviors, depending on which type of randomness it is generated from. As an application of the above results, we show that by using a filtering process based on the edge clique number, we can recover the shortest-path metric of the random geometric graph $G_mathcal{X}^*$ within a multiplicative factor of $3$, from an ER-perturbed observed graph $widehat{G}(p,q)$, for a significantly wider range of insertion probability $q$ than in previous work.
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