Geometric graph theory
mathematics, a geometric graph is a graph in which the vertices or edges are associated with geometric objects or configurations. Geometric graph theory is a specialization of graph theorythat studies geometric graphs. Notable geometric graphs and geometric graph theory problems include the following.
* A "
planar straight line graph" is a graph in which the vertices are embedded as points in the Euclidean plane, and the edges are embedded as non-crossing line segments. Fáry's theoremstates that any planar graphmay be represented as a planar straight line graph. A triangulation is a planar straight line graph to which no more edges may be added; a special case of this is the Delaunay triangulation, a graph defined from a set of points in the plane by connecting two points with an edge whenever there exists a circle containing only those two points.
* The 1-skeleton of a
polyhedronor polytopeis the set of vertices and edges of the polytope. The skeleton of any convex polyhedron is a planar graph, and the skeleton of any "k"-dimensional convex polytope is a "k"-connected graph. Conversely, Ernst Steinitzproved that any 3-connected planar graph is the skeleton of a convex polyhedron.
* A "Euclidean graph" is a graph in which the vertices represent points in the plane, and the edges are assigned lengths equal to the Euclidean distance between those points. The
Euclidean minimum spanning treeis the minimum spanning treeof a Euclidean complete graph. It is also possible to define graphs by conditions on the distances; in particular, a unit distance graphis formed by connecting pairs of points that are a unit distance apart in the plane. The Hadwiger–Nelson problemconcerns the chromatic numberof these graphs.
intersection graphis a graph in which each vertex is associated with a set and in which vertices are connected by edges whenever the corresponding sets have a nonempty intersection. When the sets are geometric objects, the result is a geometric graph. For instance, the intersection graph of line segments in one dimension is an interval graph; the intersection graph of unit disks in the plane is a unit disk graph. The Circle packing theoremstates that the intersection graphs of non-crossing circles are exactly the planar graphs. Scheinerman's conjecturestates that every planar graph can be represented as the intersection graph of line segments in the plane.
Levi graphof a family of points and lines has a vertex for each of these objects and an edge for every incident point-line pair. The Levi graphs of projective configurations lead to many important symmetric graphs and cages.
visibility graphof a closed polygon connects each pair of vertices by an edge whenever the line segment connecting the vertices lies entirely in the polygon. It is not known how to test efficiently whether an undirected graph can be represented as a visibility graph.
partial cubeis a graph for which the vertices can be associated with the vertices of a hypercube, in such a way that distance in the graph equals Hamming distancebetween the corresponding hypercube vertices. Many important families of combinatorial structures, such as the acyclic orientations of a graph or the adjacencies between regions in a hyperplane arrangement, can be represented as partial cube graphs. An important special case of a partial cube is the skeleton of the permutohedron, a graph in which vertices represent permutations of a set of ordered objects and edges represent swaps of objects adjacent in the order. Several other important classes of graphs including median graphs have related definitions involving metric embeddings Harv|Bandelt|Chepoi|2008|Ref=none.
flip graphis a graph formed from the triangulations of a point set, in which each vertex represents a triangulation and two triangulations are connected by an edge if they differ by the replacement of one edge for another. It is also possible to define related flip graphs for partitions into quadrilaterals or pseudotriangles, and for higher dimensional triangulations. The flip graph of triangulations of a convex polygon forms the skeleton of the associahedronor Stasheff polytope. The flip graph of regular triangulations of a point set (projections of higher dimensional convex hulls) can also be represented as a skeleton, of the so-called "secondary polytope".
Topological graph theory
coauthors= Chepoi, Victor
url = http://www.lif-sud.univ-mrs.fr/%7Echepoi/survey_cm_bis.pdf
title = Metric graph theory and geometry: a survey
journal = Contemp. Math.
pages = to appear
author = Pach, János, ed.
title = Towards a Theory of Geometric Graphs
year = 2004
publisher = Contemporary Mathematics, no. 342, American Mathematical Society
author = Pisanski, Tomaž; Randić, Milan
title = Bridges between geometry and graph theory
date = 2000
url = http://www.ijp.si/ftp/pub/preprints/ps/98/pp595.ps
booktitle = Geometry at Work: Papers in Applied Geometry
editor = Gorini, C. A. (Ed.)
location = Washington, DC
publisher = Mathematical Association of America
pages = 174–194
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