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{{Short description|Characterization of planar graphs by matroids}} ...e blue graph is a minimal cut in the red graph, and vice versa, so the two graphs are algebraic duals and have dual graphic matroids.]] ...

{{Short description|On forbidden minors in planar graphs}} [[File:Clique-sum.svg|thumb|upright=1.2|A clique-sum of two planar graphs and the Wagner graph, forming a ''K''₅-free graph.]] ...

...{{harvid|Scheinerman|1984}}|(1984)]], following earlier results that every planar graph could be represented as the intersection graph of a set of simple cur ...ble]] graphs, and {{harvtxt|West|1991}} conjectured that analogously every planar graph could be represented using four directions. If a graph is represented ...

{{Short description|Order dimension of incidences in planar graphs}} In [[graph theory]], '''Schnyder's theorem''' is a characterization of [[planar graph]]s in terms ...

{{about|graph theory|the conjectures in knot theory|Tait conjectures}} ...states that "Every [[K-vertex-connected graph|3-connected]] [[Planar graph|planar]] [[cubic graph]] has a [[Hamiltonian cycle]] (along the edges) through all ...

{{Short description|On cycle bases of planar graphs}} ...who published it in 1937. It states that a finite [[undirected graph]] is planar if and only if ...

{{Short description|On forbidden subgraphs in planar graphs}} ...s, named after [[Kazimierz Kuratowski]]. It states that a finite graph is planar if and only if it does not contain a [[Glossary of graph theory#Subgraph|su ...

{{Short description|Depth-first characterization of planar graphs}} }}.</ref> is a characterization of [[planar graph]]s based on the properties of the [[trémaux tree|depth-first search t ...

...rtite graph]]s ''K_m,n'' where ''m'' = ''n'', the density of the graphs approaches 1/2 as the number of vertices approaches infinity, and so the fa ...parse graph is ill-defined, and is often represented by 'roughly equal to' statements. Due to this, the way that density is defined often depends on the context ...

{{unsolved|mathematics|Are graphs uniquely determined by their subgraphs?}} Informally, the '''reconstruction conjecture''' in [[graph theory]] says that graphs are determined uniquely by their subgraphs. It is due to [[Paul Kelly (mat ...

{{About||the network protocol|Spanning Tree Protocol|other uses|}} ...is not [[connected graph|connected]] will not contain a spanning tree (see about [[#Spanning forests|spanning forests]] below). If all of the [[edge (graph ...

...alently an acyclic undirected graph, or equivalently a [[Disjoint union of graphs|disjoint union]] of trees.{{sfn|Bender|Williamson|2010|p=172}} ...var|G}} has finitely many vertices, say {{mvar|n}} of them, then the above statements are also equivalent to any of the following conditions: ...

{{short description|Planar maps require at most four colors}} ...the closures of at least three regions. Theorem: The regions of any simple planar map can be colored with only four colors, in such a way that any two adjace ...

...en. Because of this, many theorems about closed sets are dual to theorems about open sets. For example, any union of open sets is open, so dually, any inte ...example, the dual of a vector space is again a vector space. Many duality statements are not of this kind. Instead, such dualities reveal a close relation betwe ...

{{harvp|Oxley|1992}} is a standard source for basic definitions and results about matroids; {{harvp|Welsh|1976}} is an older standard source. ...cause the circuits of [[graphic matroid]]s are cycles in the corresponding graphs.<ref name=w7-9/> ...

...ints in the plane may be more easily understood in an equivalent dual form about arrangements of lines. For instance, the [[Sylvester–Gallai theorem]], stat ...ite families of [[cubic graph|3-regular]] partial cubes, isomorphic to the graphs of [[zonohedron|simple zonohedra]].{{sfnp|Eppstein|2006}} ...

...e use of [[projective duality]], in which the roles of points and lines in statements of projective geometry can be exchanged for each other. Under projective du ...uses projective duality to convert the problem into an equivalent question about arrangements of lines, which can be answered using [[Euler's polyhedral for ...

| [[Edsger Dijkstra]] invented an efficient algorithm for shortest paths in graphs as a demonstration of the abilities of the [[ARMAC]] computer. The example ...re the first semiconductor oxide(SiO2) field effect transistors; the first planar transistors, in which drain and source were adjacent at the same surface.<r ...

This is a list of notable [[theorem]]s. Lists of theorems and similar statements include: *[[Dirac's theorem on chordal graphs|Dirac's theorems]] (''[[graph theory]]'') ...

...on typically holds well with all fluids at low [[Mach number]]s (say up to about Mach 0.3), such as for modelling air winds at normal temperatures.<ref name ...he role of a conservation equation. This would seem to refute the frequent statements that the incompressible pressure enforces the divergence-free condition. ...