Plane and Planarity Thresholds for Random Geometric Graphs
نویسندگان
چکیده
A random geometric graph, G(n, r), is formed by choosing n points independently and uniformly at random in a unit square; two points are connected by a straight-line edge if they are at Euclidean distance at most r. For a given constant k, we show that n −k 2k−2 is a distance threshold function for G(n, r) to have a connected subgraph on k points. Based on that, we show that n−2/3 is a distance threshold function for G(n, r) to be plane, and n−5/8 is a distance threshold function for G(n, r) to be planar. We also investigate distance thresholds for G(n, r) to have a non-crossing edge, a clique of a given size, and an independent set of a given size.
منابع مشابه
On Sharp Thresholds in Random Geometric Graphs
We give a characterization of vertex-monotone properties with sharp thresholds in a Poisson random geometric graph or hypergraph. As an application we show that a geometric model of random k-SAT exhibits a sharp threshold for satisfiability.
متن کاملThe distant-2 chromatic number of random proximity and random geometric graphs
We are interested in finding bounds for the distant-2 chromatic number of geometric graphs drawn from different models. We consider two undirected models of random graphs: random geometric graphs and random proximity graphs for which sharp connectivity thresholds have been shown. We are interested in a.a.s. connected graphs close just above the connectivity threshold. For such subfamilies of ra...
متن کاملMonotone properties of random geometric graphs have sharp thresholds
Random geometric graphs result from taking n uniformly distributed points in the unit cube, [0, 1]d, and connecting two points if their Euclidean distance is at most r, for some prescribed r. We show that monotone properties for this class of graphs have sharp thresholds by reducing the problem to bounding the bottleneck matching on two sets of n points distributed uniformly in [0, 1]d. We pres...
متن کاملMonotone Properties of Random Geometric Graphs Have Sharp Thresholds By
Random geometric graphs result from taking n uniformly distributed points in the unit cube, [0,1]d , and connecting two points if their Euclidean distance is at most r , for some prescribed r . We show that monotone properties for this class of graphs have sharp thresholds by reducing the problem to bounding the bottleneck matching on two sets of n points distributed uniformly in [0,1]d . We pr...
متن کاملOn the Upward Planarity of Mixed Plane Graphs
A mixed plane graph is a plane graph whose edge set is partitioned into a set of directed edges and a set of undirected edges. An orientation of a mixed plane graph G is an assignment of directions to the undirected edges of G resulting in a directed plane graph G. In this paper, we study the computational complexity of testing whether a given mixed plane graph G is upward planar, i.e., whether...
متن کامل