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Forbidden graph characterizations may be used in algorithms for testing whether a graph belongs to a given family. In many cases, it is possible to test in polynomial time whether a given graph contains any of the members of the obstruction set, and therefore whether it belongs to the family defined by that obstruction set.
A closely related result, Wagner's theorem, characterizes the planar graphs by their minors in terms of the same two forbidden graphs and ,. Every Kuratowski subgraph is a special case of a minor of the same type, and while the reverse is not true, it is not difficult to find a Kuratowski subgraph (of one type or the other) from one of these ...
Pages in category "Theorems in graph theory" The following 54 pages are in this category, out of 54 total. ... Kőnig's theorem (graph theory) Kotzig's theorem ...
Alspach's theorem (graph theory) Amitsur–Levitzki theorem (linear algebra) Analyst's traveling salesman theorem (discrete mathematics) Analytic Fredholm theorem (functional analysis) Anderson's theorem (real analysis) Andreotti–Frankel theorem (algebraic geometry) Angle bisector theorem (Euclidean geometry) Ankeny–Artin–Chowla theorem ...
In the study of graph algorithms, Courcelle's theorem is the statement that every graph property definable in the monadic second-order logic of graphs can be decided in linear time on graphs of bounded treewidth. [1] [2] [3] The result was first proved by Bruno Courcelle in 1990 [4] and independently rediscovered by Borie, Parker & Tovey (1992 ...
Since such graphs have a unique embedding (up to flipping and the choice of the external face), the next bigger graph, if still planar, must be a refinement of the former graph. This allows to reduce the planarity test to just testing for each step whether the next added edge has both ends in the external face of the current embedding.
The De Bruijn–Erdős theorem for countable graphs can also be shown to be equivalent in axiomatic power, within a certain theory of second-order arithmetic, to Weak Kőnig's lemma. [ 16 ] For a counterexample to the theorem in models of set theory without choice, let G {\displaystyle G} be an infinite graph in which the vertices represent all ...
The discharging method is used to prove that every graph in a certain class contains some subgraph from a specified list. The presence of the desired subgraph is then often used to prove a coloring result. [1] Most commonly, discharging is applied to planar graphs. Initially, a charge is assigned to each face and each vertex of the graph. The ...