Planar graph
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| Planar graph | |
|---|---|
| Name | Planar graph |
| Type | Graph theory concept |
| Field | Mathematics |
Planar graph A planar graph is a graph that can be drawn on a plane without edge crossings, fundamental in Leonhard Euler's work and connected to results by Kazimierz Kuratowski and Kurt Wagner. It underpins developments tied to the Four-Color Theorem, the Kuratowski's theorem proof lineage, and algorithmic studies influenced by institutions such as the Stanford University and Massachusetts Institute of Technology. Planar graph theory interfaces with classical problems studied at venues like the International Congress of Mathematicians and has influenced computational projects at organizations including Bell Labs and IBM Research.
Definition and basic properties
A planar graph is defined by the existence of a crossing-free drawing in the plane; basic properties trace to results by Leonhard Euler and later formalizations by Gerardus 't Hooft collaborators and researchers at Princeton University. Fundamental numerical constraints relate numbers of vertices, edges, and faces, with limits used in studies by Paul Erdős and Richard P. Stanley. Planarity is invariant under homeomorphisms studied in work by Henri Poincaré and appears in combinatorial treatments by Pierre de Fermat-inspired enumerative techniques employed by researchers at University of Cambridge and University of Oxford.
Examples and non-planar graphs
Standard examples of planar graphs include the cycle graphs often analyzed in seminars at Courant Institute and tree structures used in projects at California Institute of Technology. Notable non-planar graphs central to characterization theorems are the complete graph K5 and the complete bipartite graph K3,3, which appear in classical expositions by Kazimierz Kuratowski and survey articles in journals associated with American Mathematical Society and London Mathematical Society. Historical counterexamples and constructions appear in notes by Augustin-Jean Fresnel and in problem lists circulated at conferences such as the International Mathematical Olympiad and European Congress of Mathematics.
Kuratowski's and Wagner's theorems
Kuratowski's theorem, proved by Kazimierz Kuratowski, characterizes non-planar graphs via subdivisions of K5 and K3,3 and has been reinterpreted in algorithmic contexts at University of Illinois Urbana-Champaign and by researchers affiliated with École Normale Supérieure. Wagner's theorem, attributed to Kurt Wagner, gives an equivalent formulation in terms of graph minors and connects to the broad Graph Minor Theorem program advanced by teams at Université de Paris and institutions like Microsoft Research. These theorems are pivotal in topological graph theory discussions at the International Congress of Mathematicians and influenced structural graph theory developments by Noga Alon and Endre Szemerédi.
Embeddings, faces, and Euler's formula
An embedding of a graph on the plane yields faces whose combinatorial structure is governed by Leonhard Euler's formula V − E + F = 2, a relation spotlighted in lectures at University of Göttingen and Harvard University. Dual graphs arising from embeddings connect to planar duality presentations in seminars by William Tutte and to matroid theory topics explored at Rutgers University. Studies of embeddings on surfaces generalize to toroidal and higher-genus contexts discussed at Max Planck Institute for Mathematics and in treatises by J. H. Conway and W. Thurston.
Planar graph algorithms and complexity
Planarity testing algorithms, including linear-time methods developed by researchers at Bell Labs and formalized in courses at Carnegie Mellon University, determine embeddings and support graph drawing systems from laboratories at Bell Labs and AT&T Research. Shortest-path, separator, and coloring algorithms on planar graphs have been advanced in complexity theory work at MIT and Stanford University; separator theorems by Richard J. Lipton and Robert E. Tarjan underpin divide-and-conquer schemes used in implementations by Google engineering teams. The field intersects parameterized complexity and approximation algorithms studied at University of Toronto and in workshops hosted by ACM SIGACT.
Applications and variations
Applications span circuit layout problems tackled by teams at Intel Corporation and Texas Instruments, geographic information systems projects at Esri and National Aeronautics and Space Administration, and mesh generation work in collaborations with Siemens and Argonne National Laboratory. Variations include planar embeddings on surfaces related to Bernhard Riemann's function-theoretic ideas, outerplanar and series-parallel graphs examined in industrial research by Bell Labs, and constrained-planarity cases used in software from Microsoft and Oracle Corporation. Interdisciplinary uses appear in computational biology collaborations with Broad Institute and network visualization work at Visualization Science Group.