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A map graph is a graph formed from a set of finitely many simply-connected interior-disjoint regions in the plane by connecting two regions when they share at least one boundary point. When at most three regions meet at a point, the result is a planar graph, but when four or more regions meet at a point, the result can be nonplanar (for example ...
In graph theory, a branch of mathematics, a map graph is an undirected graph formed as the intersection graph of finitely many simply connected and internally disjoint regions of the Euclidean plane. The map graphs include the planar graphs , but are more general.
In graph-theoretic terms, the theorem states that for loopless planar graph, its chromatic number is ().. The intuitive statement of the four color theorem – "given any separation of a plane into contiguous regions, the regions can be colored using at most four colors so that no two adjacent regions have the same color" – needs to be interpreted appropriately to be correct.
Let be a metric space with distance function .Let be a set of indices and let () be a tuple (indexed collection) of nonempty subsets (the sites) in the space .The Voronoi cell, or Voronoi region, , associated with the site is the set of all points in whose distance to is not greater than their distance to the other sites , where is any index different from .
The set of region allows to create the region automaton, which is a directed graph in which each node is a region, and each edge ′ ensure that ′ is a possible future of . Taking a product of this region automaton and of a timed automaton A {\displaystyle {\mathcal {A}}} which accepts a language L {\displaystyle L} creates a finite automaton ...
Likewise, graph theory is useful in biology and conservation efforts where a vertex can represent regions where certain species exist (or inhabit) and the edges represent migration paths or movement between the regions. This information is important when looking at breeding patterns or tracking the spread of disease, parasites or how changes to ...
First of all, one associates a simple planar graph to the given map, namely one puts a vertex in each region of the map, then connects two vertices with an edge if and only if the corresponding regions share a common border. The problem is then translated into a graph coloring problem: one has to paint the vertices of the graph so that no edge ...
A spatial network can be represented by a Voronoi diagram, which is a way of dividing space into a number of regions. The dual graph for a Voronoi diagram corresponds to the Delaunay triangulation for the same set of points. Voronoi tessellations are interesting for spatial networks in the sense that they provide a natural representation model ...