outerplanar graph
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| outerplanar graph | |
|---|---|
| Name | outerplanar graph |
| Family | Graph theory |
outerplanar graph An outerplanar graph is a finite simple graph that can be embedded in the plane so that all vertices lie on the boundary of its outer face. It appears in the study of William Tutte, Paul Erdős, Pál Erdős-style extremal problems, and applications in Alan Turing-era network design; it relates to classical results from Kurt Gödel, Émile Borel, and modern work by László Lovász and Noga Alon. Outerplanar graphs sit between trees and planar graphs, connecting to results involving Graph minors project contributors like Neil Robertson and Paul D. Seymour.
Definition and basic properties
An outerplanar graph is defined as a graph admitting a plane embedding with every vertex on the outer face; this notion was studied in contexts involving Kőnig's theorem applications and constructions reminiscent of work by Kazimierz Kuratowski and Kurt Wagner. Basic properties include having at most 2n−3 edges for an n-vertex biconnected outerplanar graph, being 2-vertex-colorable constraints analogous to observations of Claude Shannon-style bounds, and every outerplanar graph having degeneracy at most two, a property used by researchers such as Paul Erdős and Ronald Graham. Maximal outerplanar graphs correspond to triangulated polygons and can be characterized by a Hamiltonian cycle property studied in combinatorial traditions linking to Richard Stanley and George Pólya.
Characterizations and forbidden minors
Characterizations of outerplanar graphs include forbidden-minor descriptions: a graph is outerplanar iff it contains neither K5 nor K3,3 as a subdivision when combined with vertex-boundary constraints first analyzed in the tradition of Kuratowski's theorem; more precise forbidden-minor characterizations assert that outerplanar graphs are exactly those with no K4 and no K2,3 minor, points highlighted by contributions from Neil Robertson and Paul D. Seymour. Alternate characterizations use ear decompositions and block-cut trees appearing in algorithmic studies by scholars such as Michael Garey and David Johnson; biconnected maximal outerplanar graphs correspond to triangulations of a simple polygon, a viewpoint exploited by Donald Knuth and Herbert Wilf in enumerative contexts.
Extremal and structural results
Extremal bounds for outerplanar graphs include tight edge counts (≤ 2n−3 for n≥2) and results on vertex-face incidences used by combinatorialists like Paul Erdős and László Lovász. Structural theorems describe decomposition into series-parallel components linked with work by Seymour's decomposition collaborators and connections to Robert Tarjan's SPQR-tree ideas; treewidth of outerplanar graphs is at most two, a fact leveraged in fixed-parameter research involving Noga Alon and Fabian V. Fomin. Enumeration results for maximal outerplanar graphs relate to Catalan-number families studied by Eugène Charles Catalan admirers and analytic combinatorics approaches of Philippe Flajolet.
Algorithms and computational complexity
Recognition of outerplanar graphs can be performed in linear time using DFS and planarity testing techniques inspired by algorithms of John Hopcroft and Robert Tarjan. Many optimization problems that are hard on general graphs become tractable on outerplanar graphs: coloring, Hamiltonicity checks for maximal outerplanar instances, and shortest-path problems exploit small treewidth and dynamic programming paradigms popularized by Richard Karp and Jure Leskovec-adjacent algorithmic traditions. Problems like maximum independent set and graph isomorphism admit efficient algorithms on outerplanar instances, reflecting methodologies from Leslie Valiant and László Lovász in algebraic and combinatorial algorithm design.
Applications and examples
Outerplanar graphs model ring networks and polygonal triangulations used in computational geometry and network routing problems studied by John Conway-adjacent computational researchers and practitioners at institutions like Bell Labs and IBM Research. Examples include trees, cycles, and series-parallel graphs arising in circuit design research connected to pioneers like Claude Shannon and Gordon Moore. In geographic information systems and cartography projects associated with organizations such as National Geographic Society and Ordnance Survey, outerplanar structures model boundary-constrained connectivity; they also appear in molecule-shape abstractions referenced in chemical graph theory literature influenced by August Kekulé and Linus Pauling.
Variations and related graph classes
Related classes include outerplanar-like families such as k-outerplanar graphs studied in parameterized complexity by researchers like Rod Downey and Michael Fellows, series-parallel graphs connected to Seymour's work and Robert Tarjan's algorithms, and planar graphs at large as in the work of William Tutte and Kurt Wagner. Other variants include weakly outerplanar and near-outerplanar classes used in approximation schemes inspired by contributions from David S. Johnson and Christos Papadimitriou; connections to map graphs and Halin graphs echo themes from George B. Dantzig-style optimization and combinatorial topology studied by Raoul Bott.