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Theorem relating graph minors and topological embeddings
In mathematics, the graph structure theorem is a major result in the area of graph theory. The result establishes a deep and fundamental connection between
Graph_structure_theorem
Finiteness of sets of forbidden graph minors
graph theory, the Robertson–Seymour theorem (also called the graph minors theorem) states that the undirected graphs, partially ordered by the graph minor
Robertson–Seymour_theorem
Subgraph with contracted edges
conjectures involving graph minors include the graph structure theorem, according to which the graphs that do not have H as a minor may be formed by gluing
Graph_minor
On forbidden minors in planar graphs
In graph theory, Wagner's theorem is a mathematical forbidden graph characterization of planar graphs, named after Klaus Wagner, stating that a finite
Wagner's_theorem
(combinatorics) Graph structure theorem (graph theory) Grinberg's theorem (graph theory) Grötzsch's theorem (graph theory) Hajnal–Szemerédi theorem (graph theory)
List_of_theorems
Area of discrete mathematics
computer science, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context
Graph_theory
Planar maps require at most four colors
a graph coloring of the planar graph of adjacencies between regions. In graph-theoretic terms, the theorem states that for a loopless planar graph G {\displaystyle
Four_color_theorem
On bipartite matching and vertex cover
In the mathematical area of graph theory, Kőnig's theorem, proved by Dénes Kőnig (1931), describes an equivalence between the maximum matching problem
Kőnig's theorem (graph theory)
Kőnig's_theorem_(graph_theory)
Basic concept of graph theory
facts about connectivity in graphs is Menger's theorem, which characterizes the connectivity and edge-connectivity of a graph in terms of the number of
Connectivity_(graph_theory)
Theorems connecting continuity to closure of graphs
closed graph theorem is a result connecting the continuity of a linear operator to a topological property of their graph. Precisely, the theorem states
Closed graph theorem (functional analysis)
Closed_graph_theorem_(functional_analysis)
Graph that can be embedded in the plane
consequence, planar graphs also have treewidth and branch-width O(√n). The planar product structure theorem states that every planar graph is a subgraph of
Planar_graph
On linear-time algorithms for graph logic
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
Courcelle's_theorem
Path in a graph that visits each vertex exactly once
Ore's theorems basically state that a graph is Hamiltonian if it has enough edges. The Bondy–Chvátal theorem operates on the closure cl(G) of a graph G with
Hamiltonian_path
Property of artificial neural networks
machine learning, the universal approximation theorems (UATs) state that neural networks with a certain structure can, in principle, approximate any continuous
Universal approximation theorem
Universal_approximation_theorem
Statement in mathematical combinatorics
In combinatorics, Ramsey's theorem, in one of its graph-theoretic forms, states that one will find monochromatic cliques in any edge labelling (with colours)
Ramsey's_theorem
Bijection between the vertex set of two graphs
isomorphic but both have K3 as their line graph. The Whitney graph theorem can be extended to hypergraphs. While graph isomorphism may be studied in a classical
Graph_isomorphism
Graph defined from a mathematical group
Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group, is a graph that encodes the abstract structure of a group
Cayley_graph
On chains and antichains in partial orders
Dilworth's theorem is equivalent to Kőnig's theorem on bipartite graph matching and several other related theorems including Hall's marriage theorem. To prove
Dilworth's_theorem
Gluing graphs at complete subgraphs
graphs with the eight-vertex Wagner graph; this structure theorem can be used to show that the four color theorem is equivalent to the case k = 5 of the
Clique-sum
On coloring the edges of graphs
In graph theory, Vizing's theorem states that every simple undirected graph may be edge colored using a number of colors that is at most one larger than
Vizing's_theorem
Theorem about infinite graphs
In graph theory, a branch of mathematics, Halin's grid theorem states that the infinite graphs with thick ends are exactly the graphs containing subdivisions
Halin's_grid_theorem
Theorem about a certain class of control-flow graphs
language theory, the structured program theorem, generally called the Böhm–Jacopini theorem, states that a class of control-flow graphs (historically called
Structured_program_theorem
Graph-theoretic description of polyhedra
combinatorics, a branch of mathematics, Steinitz's theorem is a characterization of the undirected graphs formed by the edges and vertices of three-dimensional
Steinitz's_theorem
Describing a family of graphs by excluding certain (sub)graphs
forbidden graphs, the complete graph K5 and the complete bipartite graph K3,3. For Kuratowski's theorem, the notion of containment is that of graph homeomorphism
Forbidden graph characterization
Forbidden_graph_characterization
Branch of the mathematical field of graph theory
circuit boards. Graph embeddings are also used to prove structural results about graphs, via graph minor theory and the graph structure theorem. Crossing number
Topological_graph_theory
Graph representing edges of another graph
underlying graph from vertices into edges, and by Whitney's theorem the same translation can also be done in the other direction. Line graphs are claw-free
Line_graph
Graph with tight clique-coloring relation
graph theorem states that the complement graph of a perfect graph is also perfect. The strong perfect graph theorem characterizes the perfect graphs in
Perfect_graph
Characterizes the height of any finite partially ordered set
to Dilworth's theorem on the widths of partial orders, to the perfection of comparability graphs, to the Gallai–Hasse–Roy–Vitaver theorem relating longest
Mirsky's_theorem
Theorem in extremal graph theory
extremal graph theory, the Erdős–Stone theorem is an asymptotic result generalising Turán's theorem to bound the number of edges in an H-free graph for a
Erdős–Stone_theorem
Graph divided into two independent sets
In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets
Bipartite_graph
Adjacent subset of an undirected graph
edges must contain a three-vertex clique. Ramsey's theorem states that every graph or its complement graph contains a clique with at least a logarithmic number
Clique_(graph_theory)
On coloring infinite graphs
In graph theory, the De Bruijn–Erdős theorem relates graph coloring of an infinite graph to the same problem on its finite subgraphs. It states that,
De Bruijn–Erdős theorem (graph theory)
De_Bruijn–Erdős_theorem_(graph_theory)
Bivariegated graph Cage (graph theory) Cayley graph Circle graph Clique graph Cograph Common graph Complement of a graph Complete graph Cubic graph Cycle graph De
List_of_graph_theory_topics
Appendix:Glossary of graph theory in Wiktionary, the free dictionary. This is a glossary of graph theory. Graph theory is the study of graphs, systems of nodes
Glossary_of_graph_theory
Set of edges without common vertices
bipartite graphs. Hall's marriage theorem provides a characterization of bipartite graphs which have a perfect matching and Tutte's theorem on perfect
Matching_(graph_theory)
On tangency patterns of circles
packing theorem applies to any polyhedral graph and its dual graph, and proves the existence of a primal–dual packing, circle packings for both graphs that
Circle_packing_theorem
Graph with all vertices of degree 3
of graph theory, a cubic graph is a graph in which all vertices have degree three. In other words, a cubic graph is a 3-regular graph. Cubic graphs are
Cubic_graph
Structure-preserving correspondence between node-link graphs
the mathematical field of graph theory, a graph homomorphism is a mapping between two graphs that respects their structure. More concretely, it is a function
Graph_homomorphism
Type of knowledge base
knowledge graph is a knowledge base that uses a graph-structured data model or topology to represent and operate on data. Knowledge graphs are often used
Knowledge_graph
Graph whose embedding in a Euclidean space forms a regular tiling
In graph theory, a lattice graph, mesh graph, or grid graph is a graph whose drawing, embedded in some Euclidean space R n {\displaystyle \mathbb {R}
Lattice_graph
Mathematical graph theorem
mathematical discipline of graph theory, Petersen's theorem, named after Julius Petersen, is one of the earliest results in graph theory and can be stated
Petersen's_theorem
Methodic assignment of colors to elements of a graph
graph introduced by Shannon. The conjecture remained unresolved for 40 years, until it was established as the celebrated strong perfect graph theorem
Graph_coloring
Equivalence between strongly orientable graphs and bridgeless graphs
In graph theory, Robbins' theorem, named after Herbert Robbins (1939), states that the graphs that have strong orientations are exactly the 2-edge-connected
Robbins'_theorem
Graphs of d-dimensional polytopes are d-connected
combinatorics, a branch of mathematics, Balinski's theorem is a statement about the graph-theoretic structure of three-dimensional convex polyhedra and higher-dimensional
Balinski's_theorem
Branch of mathematical combinatorics
the density version of the Hales-Jewett theorem. Ergodic Ramsey theory Extremal graph theory Goodstein's theorem Bartel Leendert van der Waerden Discrepancy
Ramsey_theory
Formula for number of orbits of a group action
cycle structure of the action of the group elements; see here). Thus, according to the enumeration theorem, the generating function of graphs on 3 vertices
Pólya_enumeration_theorem
Graph data structure
In computer science, an e-graph is a data structure that stores an equivalence relation over terms of some language. Let Σ {\displaystyle \Sigma } be
E-graph
Triangle-free graph requiring four colors
his 1959 theorem that planar triangle-free graphs are 3-colorable. The Grötzsch graph is a member of an infinite sequence of triangle-free graphs, each the
Grötzsch_graph
Relationship between derivatives and integrals
first fundamental theorem may be interpreted as follows. Given a continuous function y = f ( x ) {\displaystyle y=f(x)} whose graph is plotted as a curve
Fundamental theorem of calculus
Fundamental_theorem_of_calculus
Trail in a graph that visits each edge once
Euler's Theorem: A connected graph has an Euler cycle if and only if every vertex has an even number of incident edges. The term Eulerian graph has two
Eulerian_path
Maximal subgraph whose vertices can reach each other
Numbers of components play a key role in Tutte's theorem on perfect matchings characterizing finite graphs that have perfect matchings and the associated
Component_(graph_theory)
Equivalence of distributive lattices and set families
similarly named results, see Birkhoff's theorem (disambiguation). In mathematics, Birkhoff's representation theorem for distributive lattices states that
Birkhoff's representation theorem
Birkhoff's_representation_theorem
Undirected, connected, and acyclic graph
undirected graph is a forest. The various kinds of data structures referred to as trees in computer science have underlying graphs that are trees in graph theory
Tree_(graph_theory)
Graph which remains connected when k or fewer nodes removed
Menger's theorem (Diestel 2005, p. 55). This definition produces the same answer, n − 1, for the connectivity of the complete graph Kn. A k-connected graph is
Vertex_connectivity
Any planar graph can be subdivided by removing a few vertices
In graph theory, the planar separator theorem is a form of isoperimetric inequality for planar graphs, that states that any planar graph can be split
Planar_separator_theorem
3-regular graph with no 3-edge-coloring
four color theorem is that every snark is a non-planar graph. Research on snarks originated in Peter G. Tait's work on the four color theorem in 1880, but
Snark_(graph_theory)
Graph representing faces of another graph
mathematical discipline of graph theory, the dual graph of a planar graph G is a graph that has a vertex for each face of G. The dual graph has an edge for each
Dual_graph
Graph with sign-labeled edges
In the area of graph theory in mathematics, a signed graph is a graph in which each edge has a positive or negative sign. A signed graph is balanced if
Signed_graph
Sparse graph with strong connectivity
In graph theory, an expander graph is a sparse graph that has strong connectivity properties, quantified using vertex, edge or spectral expansion. Expander
Expander_graph
Part of the mathematical subject of group theory
known as the structure theorem. One of the immediate consequences is the classic Kurosh subgroup theorem describing the algebraic structure of subgroups
Bass–Serre_theory
Ideals in a Boolean algebra can be extended to prime ideals
is known as the ultrafilter lemma. Other theorems are obtained by considering different mathematical structures with appropriate notions of ideals, for
Boolean_prime_ideal_theorem
be no Hamiltonian cycle. The resulting graph is 3-connected and planar, so by Steinitz' theorem it is the graph of a polyhedron. It has 25 faces. It can
Tutte_graph
vertex set of a set of edges E {\displaystyle E} . Geiringer-Laman Theorem. A graph G = ( V , E ) {\displaystyle G=(V,E)} is generically rigid in 2 {\displaystyle
Geiringer–Laman_theorem
Study of graphs defined by geometric means
Fáry's theorem states that any planar graph may be represented as a planar straight line graph. A triangulation is a planar straight line graph to which
Geometric_graph_theory
Graph without four-vertex star subgraphs
papers in which they prove a structure theory for claw-free graphs, analogous to the graph structure theorem for minor-closed graph families proven by Robertson
Claw-free_graph
Theorem in topology
curve theorem, do not generalize to Z 2 {\displaystyle \mathbb {Z} ^{2}} under either graph structure. If the "6-neighbor square grid" structure is imposed
Jordan_curve_theorem
Graph obeys some properties of random graphs
In graph theory, a graph is said to be a pseudorandom graph if it obeys certain properties that random graphs obey with high probability. There is no concrete
Pseudorandom_graph
In mathematics, specifically the 2-category theory, the pasting theorem states that every 2-categorical pasting scheme defines a unique composite 2-cell
Pasting_theorem
Graph with oriented edges
In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed
Directed_graph
Logical formulation of graph properties
include random graphs, interval graphs, and (through a logical expression of the graph structure theorem) every class of graphs characterized by forbidden
Logic_of_graphs
Upper bound on intersecting set families
{\displaystyle n\geq 2r} . The theorem may also be formulated in terms of graph theory: the independence number of the Kneser graph K G n , r {\displaystyle
Erdős–Ko–Rado_theorem
Field of mathematics which studies incidence structures
Even with this severe limitation, theorems can be proved and interesting facts emerge concerning this structure. Such fundamental results remain valid
Incidence_geometry
Canadian-American mathematician (born 1938)
of this work, Robertson and Seymour also proved the graph structure theorem describing the graphs in these families. Additional major results in Robertson's
Neil Robertson (mathematician)
Neil_Robertson_(mathematician)
Unproven generalization of the four-color theorem
k = 5 {\displaystyle k=5} implies the four color theorem: for, if the conjecture is true, every graph requiring five or more colors would have a K 5 {\displaystyle
Hadwiger conjecture (graph theory)
Hadwiger_conjecture_(graph_theory)
Operation combining two oriented knots
opposite colors. The Jordan curve theorem implies that there is exactly one such coloring. We construct a new plane graph whose vertices are the white faces
Knot_(mathematics)
Infinite graph containing all countable graphs
In the mathematical field of graph theory, the Rado graph, Erdős–Rényi graph, or random graph is a countably infinite graph that can be constructed (with
Rado_graph
countable graph have an unfriendly partition into two parts? Vizing's conjecture on the domination number of cartesian products of graphs Walescki's theorem for
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Directed graph where each vertex pair has one arc
Rédei's theorem is the special case for complete graphs of the Gallai–Hasse–Roy–Vitaver theorem, relating the lengths of paths in orientations of graphs to
Tournament_(graph_theory)
Directed path algebra
gauge-invariant uniqueness theorem and Cuntz-Krieger uniqueness theorem for graph C*-algebras. Formal statements of the uniqueness theorems are as follows: The
Leavitt_path_algebra
Branch of mathematics
1, 1, 1, 3). Several theorems relate properties of the spectrum to other graph properties. As a simple example, a connected graph with diameter D will
Algebraic_graph_theory
Duality of graph colorings and orientations
In graph theory, the Gallai–Hasse–Roy–Vitaver theorem is a form of duality between the colorings of the vertices of a given undirected graph and the orientations
Gallai–Hasse–Roy–Vitaver theorem
Gallai–Hasse–Roy–Vitaver_theorem
Representation of a graph as a path graph "thickened" by some amount
minors for pathwidth-2 graphs has been computed; it contains 110 different graphs. The graph structure theorem for minor-closed graph families states that
Pathwidth
Topics referred to by the same term
configuration, a geometric configuration related to 'Pappus's theorem' Pappus graph, a graph related to the pappus configuration Papus (disambiguation) Pappu
Pappus
Optimization technique
solving a maximum flow problem in a graph (and thus, by the max-flow min-cut theorem, define a minimal cut of the graph). Under most formulations of such
Graph cuts in computer vision and artificial intelligence
Graph_cuts_in_computer_vision_and_artificial_intelligence
vertices. the edge graphs of 3-dimensional polytopes are rich in structure but well-understood: by Steinitz's theorem the edge graphs of 3-polytopes are
Graph_of_a_polytope
for Turán's theorem come from the Turán graph T ( n , r − 1 ) {\displaystyle T(n,r-1)} . This result can be generalized to arbitrary graphs G {\displaystyle
Forbidden_subgraph_problem
Graph which can be made planar by removing a single node
Apex-minor-free graph families obey a strengthened version of the graph structure theorem, leading to additional approximation algorithms for graph coloring
Apex_graph
Combinatorial theory of mechanics and discrete geometry
Frameworks Kempe's universality theorem Weisstein, Eric W. "Rigid Graph". MathWorld. Weisstein, Eric W. "Flexible Graph". MathWorld. Chen, L. (2022), "Triangular
Structural_rigidity
theory is a subfield of extremal graph theory. It studies common generalizations of Ramsey's theorem and Turán's theorem. In brief, Ramsey-Turán theory
Ramsey-Turán_theory
Directed graph where every node has exactly one path to it from the root
In graph theory, an arborescence is a directed graph where there exists a vertex r (called the root) such that, for any other vertex v, there is exactly
Arborescence_(graph_theory)
formulate theorems that apply simultaneously to all of these subclasses and contain specific results for each subclass as special cases. Although graph C*-algebras
Graph_C*-algebra
Graph without triples of adjacent vertices
defined as graphs with clique number ≤ 2, graphs with girth ≥ 4, graphs with no induced 3-cycle, or locally independent graphs. By Turán's theorem, the n-vertex
Triangle-free_graph
Every graph has evenly many odd vertices
In graph theory, the handshaking lemma is the statement that, in every finite undirected graph, the number of vertices that touch an odd number of edges
Handshaking_lemma
Branch of geometry that studies combinatorial properties and constructive methods
Buildings An oriented matroid is a mathematical structure that abstracts the properties of directed graphs and of arrangements of vectors in a vector space
Discrete_geometry
Number of edges touching a vertex in a graph
By Brooks' theorem, any graph G other than a clique or an odd cycle has chromatic number at most Δ(G), and by Vizing's theorem any graph has chromatic
Degree_(graph_theory)
Assigning directions to the edges of an undirected graph
Nešetřil, Jaroslav; Ossona de Mendez, Patrice (2012), "Theorem 3.13", Sparsity: Graphs, Structures, and Algorithms, Algorithms and Combinatorics, vol. 28
Orientation_(graph_theory)
Graph where all long cycles have a chord
In any graph, a vertex separator is a set of vertices the removal of which leaves the remaining graph disconnected. According to a theorem of Dirac
Chordal_graph
Solid with eight equal triangular faces
octahedron give rise to a graph, a discrete structure drawn in a plane. The name is octahedral graph. The octahedral graph is an example of a four-connected
Regular_octahedron
Study of discrete mathematical structures
attention within areas of the field. In graph theory, much research was motivated by attempts to prove the four color theorem, first stated in 1852, but not proved
Discrete_mathematics
Theorem in algebra
as the fundamental group of a finite graph of finite groups. The modern formulation of the Muller–Schupp theorem is as follows: Let G be a finitely generated
Muller–Schupp_theorem
GRAPH STRUCTURE-THEOREM
GRAPH STRUCTURE-THEOREM
Boy/Male
Arabic, Modern
Grape
Girl/Female
Tamil
Shape, Structure
Girl/Female
Indian
Grape vine
Boy/Male
Afghan, Arabic, Gujarati, Indian, Muslim
Solid Structure; Lifetime
Boy/Male
Indian
Solid structure
Boy/Male
Indian
Good Structure
Girl/Female
Hindu, Indian, Telugu
The Structure of God
Girl/Female
Muslim
Grape like
Boy/Male
Muslim
Solid structure
Boy/Male
Muslim
Grape
Girl/Female
Indian
Shape, Structure
Girl/Female
Indian
Structure
Girl/Female
Muslim
Grape vine
Girl/Female
Tamil
Shape, Structure
Girl/Female
Indian
Shape, Structure
Girl/Female
Indian
Grape like
Girl/Female
Indian, Kashmiri
Body Structure
Boy/Male
African, Arabic
Grape Vines
Girl/Female
Arabic, Assamese, Hindu, Indian, Kannada, Malayalam, Marathi, Muslim, Telugu
Grape
Boy/Male
Indian
Grape
GRAPH STRUCTURE-THEOREM
GRAPH STRUCTURE-THEOREM
Boy/Male
Hindu
Vishnu, Husband of Goddess Lakshmi
Male
English
English surname transferred to forename use, derived from an Old English byname, Red, READ means "red-headed or ruddy-complexioned."Â
Girl/Female
Biblical
Keeping counsel, fastened.
Girl/Female
Hindu
Lord of Yoga
Girl/Female
American, British, English, French, Hebrew
God is My Judge; Feminine Variant of Daniel
Girl/Female
Muslim/Islamic
Noble Honourable
Boy/Male
Tamil
Lord Vishnu
Boy/Male
Hindu, Indian, Tamil, Traditional
Refuge
Boy/Male
Tamil
Imaging of God, Lord of perfection
Girl/Female
Biblical
City, vocation, meeting.
GRAPH STRUCTURE-THEOREM
GRAPH STRUCTURE-THEOREM
GRAPH STRUCTURE-THEOREM
GRAPH STRUCTURE-THEOREM
GRAPH STRUCTURE-THEOREM
n.
Composition, or structure.
n.
A touch of adverse criticism; censure.
a.
Resembling a grape.
n.
A seed of the grape.
a.
Of or pertaining to structure; affecting structure; as, a structural error.
n.
A stroke; a glance; a touch.
n.
The act of building; the practice of erecting buildings; construction.
n.
Arrangement of parts, of organs, or of constituent particles, in a substance or body; as, the structure of a rock or a mineral; the structure of a sentence.
a.
Affected with a stricture; as, a strictured duct.
n.
That which is built; a building; esp., a building of some size or magnificence; an edifice.
a.
Of or pertaining to organit structure; as, a structural element or cell; the structural peculiarities of an animal or a plant.
n.
Manner of organization; the arrangement of the different tissues or parts of animal and vegetable organisms; as, organic structure, or the structure of animals and plants; cellular structure.
n.
A localized morbid contraction of any passage of the body. Cf. Organic stricture, and Spasmodic stricture, under Organic, and Spasmodic.
n.
Manner of building; form; make; construction.
n.
A stria.
n.
Strictness.
n.
A sort of grape.
n.
Organic structure; organization.
a.
Having a definite organic structure; showing differentiation of parts.