These tools include a dynamic graph formalism, various computational models, and communication models for distributed networks. In this report, we identify a collection of recent theoretical tools whose purpose is to model, describe, and leverage dynamic networks in a formal way. As a result, it is hard and sometimes impossible to guarantee, mathematically, that a given algorithm will reach its objectives once deployed in real conditions. Unfortunately, few theoretical tools to date have enabled the study of dynamic networks in a formal and rigorous way. This trend exists both in everyday life (e.g., smartphones, vehicles, and commercial satellites) and in a military context (e.g., dismounted soldiers or swarms of UAVs). Weisstein, Eric W., " Unit-Distance Graph", MathWorld.The number of telecommunication networks deployed in a dynamic environment is quickly growing.Unit disk graph, a graph on the plane that has an edge whenever two points are at distance at most one.It is NP-hard, and more specifically complete for the existential theory of the reals, to test whether a given graph is a unit distance graph, or is a strict unit distance graph Template:Harv. The dimension needed to embed a graph so that all edges have unit distance, and the dimension needed to embed a graph so that the edges are exactly the unit distance pairs, may greatly differ from each other: the 2 n-vertex crown graph may be embedded in four dimensions so that all its edges have unit length, but requires at least n − 2 dimensions to be embedded so that the edges are the only unit-distance pairs Template:Harv. The definition of a unit distance graph may naturally be generalized to any higher-dimensional Euclidean space.Īny graph may be embedded as a set of points in a sufficiently high dimension Template:Harvtxt show that the dimension necessary to embed a graph in this way may be bounded by twice its maximum degree. The full Beckman–Quarles theorem states that any transformation of the Euclidean plane (or a higher-dimensional space) that preserves unit distances must be a congruence that is, for the infinite unit distance graph whose vertices are all the points in the plane, any graph automorphism must be an isometry Template:Harv. This result implies a finite version of the Beckman–Quarles theorem: for any two points p and q at distance A, there exists a finite rigid unit distance graph containing p and q such that any transformation of the plane that preserves the unit distances in this graph preserves the distance between p and q Template:Harv. Representation of algebraic numbers and the Beckman–Quarles theoremįor every algebraic number A, it is possible to find a unit distance graph G in which some pair of vertices are at distance A in all unit distance representations of G Template:Harvs. This bound can also be viewed as counting incidences between points and unit circles, and is closely related to the Szemerédi–Trotter theorem on incidences between points and lines. The best known upper bound for this problem, due to Template:Harvs, is proportional to Īnd offered a prize of $500 for determining whether or not the maximum number of unit distances can also be upper bounded by a function of this form Template:Harv. The hypercube graph provides a lower bound on the number of unit distances proportional to n log n. In graph theoretic terms, how dense can a unit distance graph be? Template:Harvs posed the problem of estimating how many pairs of points in a set of n points could be at unit distance from each other. The graph formed by removing one of the spokes from the wheel graph W 7 is a subgraph of a unit distance graph, but is not a strict unit distance graph: there is only one way ( up to congruence) to place the vertices at distinct locations such that adjacent vertices are a unit distance apart, and this placement also puts the two endpoints of the missing spoke at unit distance Template:Harv.Ĭounting unit distances List of unsolved problems in mathematics How many unit distances can be determined by a set of n points? In order to distinguish the two definitions, the graphs in which non-edges are required to be a non-unit distance apart may be called strict unit distance graphs Template:Harv. For instance, the Möbius-Kantor graph has a drawing of this type.Īccording to this looser definition of a unit distance graph, all generalized Petersen graphs are unit distance graphs Template:Harv. Some sources define a graph as being a unit distance graph if its vertices can be mapped to distinct locations in the plane such that adjacent pairs are at unit distance apart, disregarding the possibility that some non-adjacent pairs might also be at unit distance apart. A unit distance drawing of the Möbius-Kantor graph in which some nonadjacent pairs are also at unit distance from each other.
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