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Fewest cliques covering a graph's edges
mathematical field of graph theory, the intersection number of a graph G = ( V , E ) {\displaystyle G=(V,E)} is the smallest number of elements needed to
Intersection number (graph theory)
Intersection_number_(graph_theory)
Graph representing intersections between given sets
In graph theory, an intersection graph is a graph that represents the pattern of intersections of a family of sets. Any graph can be represented as an
Intersection_graph
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
Fewest edge crossings in drawing of a graph
graph theory, the crossing number cr(G) of a graph G is the lowest number of edge crossings of a plane drawing of the graph G. For instance, a graph is
Crossing number (graph theory)
Crossing_number_(graph_theory)
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
Intersection graph for intervals on the real number line
intervals intersect. It is the intersection graph of the intervals. Interval graphs are chordal graphs and perfect graphs. They can be recognized in linear
Interval_graph
Procedures for constructing new graphs in graph theory
In the mathematical field of graph theory, graph operations are operations which produce new graphs from initial ones. They include both unary (one input)
Graph_operations
Linear algebra aspects of graph theory
In mathematics, spectral graph theory is the study of the properties of a graph in relationship to the characteristic polynomial, eigenvalues, and eigenvectors
Spectral_graph_theory
Influence of local substructure of a graph on global properties
Extremal graph theory is a branch of combinatorics, itself an area of mathematics, that lies at the intersection of extremal combinatorics and graph theory. In
Extremal_graph_theory
Adjacent subset of an undirected graph
In graph theory, a clique (/ˈkliːk/ or /ˈklɪk/) is a subset of vertices of an undirected graph such that every two distinct vertices in the clique are
Clique_(graph_theory)
Intersection graph of a chord diagram
In graph theory, a circle graph is the intersection graph of a chord diagram. That is, it is an undirected graph whose vertices can be associated with
Circle_graph
Intersection graph of unit intervals on the real line
In graph theory, a branch of mathematics, an indifference graph is an undirected graph constructed by assigning a real number to each vertex and connecting
Indifference_graph
Methodic assignment of colors to elements of a graph
In graph theory, graph coloring is a methodic assignment of labels traditionally called "colors" to elements of a graph. The assignment is subject to certain
Graph_coloring
Edge whose deletion would disconnect a graph
In graph theory, a bridge, isthmus, cut-edge, or cut arc is an edge of a graph whose deletion increases the graph's number of connected components. Equivalently
Bridge_(graph_theory)
Graph that can be embedded in the plane
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect
Planar_graph
Unrelated vertices in graphs
In graph theory, an independent set, stable set, coclique or anticlique is a set of vertices in a graph, no two of which are adjacent. That is, it is a
Independent set (graph theory)
Independent_set_(graph_theory)
Shared independent set of two matroids
matroid intersection problem is to find a common independent set with the maximum possible weight. These problems generalize many problems in graph theory and
Matroid_intersection
Graph representing a permutation
In the mathematical field of graph theory, a permutation graph is a graph whose vertices represent the elements of a permutation, and whose edges represent
Permutation_graph
discrete and Euclidean geometries, graph theory, group theory, mathematical logic, number theory, set theory, Ramsey theory, dynamical systems, and partial
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Approach to studying how topology affects evolution of a population
Evolutionary graph theory is an area of research lying at the intersection of graph theory, probability theory, and mathematical biology. Evolutionary graph theory
Evolutionary_graph_theory
When every path in a control-flow graph must go through one node to reach another
postdominate any other strict postdominators of n. Control-flow graph Interval (graph theory) Static single assignment form Lengauer, Thomas; Tarjan, Robert
Dominator_(graph_theory)
Graph where all long cycles have a chord
In the mathematical area of graph theory, a chordal graph is one in which all cycles of four or more vertices have a chord, which is an edge that is not
Chordal_graph
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
Study of graphs defined by geometric means
Geometric graph theory in the broader sense is a large and amorphous subfield of graph theory, concerned with graphs defined by geometric means. In a stricter
Geometric_graph_theory
of graph theory, the sphericity of a graph is a graph invariant defined to be the smallest dimension of Euclidean space required to realize the graph as
Sphericity_(graph_theory)
Set of elements common to all of some sets
In set theory, the intersection of two sets A {\displaystyle A} and B , {\displaystyle B,} denoted by A ∩ B , {\displaystyle A\cap B,} is the set containing
Intersection_(set_theory)
Graph representing edges of another graph
In the mathematical discipline of graph theory, the line graph of an undirected graph G is another graph L(G) that represents the adjacencies between edges
Line_graph
Refinement of perfect matching theorems
Deficiency is a concept in graph theory that is used to refine various theorems related to perfect matching in graphs, such as Hall's marriage theorem
Deficiency_(graph_theory)
Graph without triples of adjacent vertices
area of graph theory, a triangle-free graph is an undirected graph in which no three vertices form a triangle of edges. Triangle-free graphs may be equivalently
Triangle-free_graph
Intersection graph of unit disks in the plane
geometric graph theory, a unit disk graph is the intersection graph of a family of unit disks in the Euclidean plane. That is, it is a graph with one vertex
Unit_disk_graph
Set of edges without common vertices
In the mathematical discipline of graph theory, a matching or independent edge set in an undirected graph is a set of edges without common vertices. In
Matching_(graph_theory)
Intersection graph of trapezoids between parallel lines
In graph theory, trapezoid graphs are intersection graphs of trapezoids between two horizontal lines. They are a class of co-comparability graphs that
Trapezoid_graph
Non-crossing graph with vertices on outer face
In graph theory, an outerplanar graph is a graph that has a planar drawing for which all vertices belong to the outer face of the drawing. Outerplanar
Outerplanar_graph
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
Graph whose biconnected components are all cliques
In graph theory, a branch of combinatorial mathematics, a block graph or clique tree is a type of undirected graph in which every biconnected component
Block_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
Intersection graph for curves in the plane
graph theory, a string graph is an intersection graph of curves in the plane; each curve is called a "string". Given a graph G, G is a string graph if
String_graph
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
Method of graph decomposition
In graph theory, a bramble for an undirected graph G is a family of connected subgraphs of G that all touch each other: for every pair of disjoint subgraphs
Bramble_(graph_theory)
Graph generated by a random process
The theory of random graphs lies at the intersection between graph theory and probability theory. From a mathematical perspective, random graphs are used
Random_graph
Graph which partitions into a clique and independent set
In graph theory, a branch of mathematics, a split graph is a graph in which the vertices can be partitioned into a clique and an independent set. Split
Split_graph
In graph theory, the Games graph is the largest known locally linear strongly regular graph. Its parameters as a strongly regular graph are (729,112,1
Games_graph
Maximal subgraph whose vertices can reach each other
In graph theory, a component of an undirected graph is a connected subgraph that is not part of any larger connected subgraph. The components of any graph
Component_(graph_theory)
Generalized notion of counting curve intersections
a line, the intersection number along the line should be at least two. These questions are discussed systematically in intersection theory. Let X be a
Intersection_number
Topics referred to by the same term
crossing number is the sum of positive and negative crossings Crossing number (graph theory) of a graph is the minimal number of edge intersections in any
Crossing_number
Class of simple graphs defined from vector spaces
In graph theory, Grassmann graphs are a special class of simple graphs defined from systems of subspaces. The vertices of the Grassmann graph Jq(n, k)
Grassmann_graph
Graph whose vertices correspond to combinations of a set of n elements
In graph theory, the Kneser graph K(n, k) (alternatively KGn,k) is the graph whose vertices correspond to the k-element subsets of a set of n elements
Kneser_graph
In graph theory, a class of graphs is said to have few cliques if every member of the class has a polynomial number of maximal cliques. Certain generally
Graphs_with_few_cliques
Graph property
mathematical field of graph theory, a distance-regular graph is a regular graph such that for any two vertices v and w, the number of vertices at distance
Distance-regular_graph
Bipartite 3-regular graph with 90 vertices and 135 edges
mathematical field of graph theory, the Foster graph is a bipartite 3-regular graph with 90 vertices and 135 edges. The Foster graph is Hamiltonian and has
Foster_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
Concept in graph theory
In graph theory, a vertex is incident with an edge if the vertex is one of the two vertices the edge connects. An incidence is a pair ( u , e ) {\displaystyle
Incidence_(graph)
Study of discrete mathematical structures
also continuous graphs; however, for the most part, research in graph theory falls within the domain of discrete mathematics. Number theory is concerned
Discrete_mathematics
Graph where every edge is in one triangle
In graph theory, a locally linear graph is an undirected graph in which every edge belongs to exactly one triangle. Equivalently, for each vertex of the
Locally_linear_graph
Mathematical tree of cycles
In graph theory, a cactus (sometimes called a cactus tree) is a connected graph in which any two simple cycles have at most one vertex in common. Equivalently
Cactus_graph
Graph whose induced subgraphs preserve distance
In graph theory, a branch of discrete mathematics, a distance-hereditary graph (also called a completely separable graph) is a graph in which the distances
Distance-hereditary_graph
All even-degree subgraphs of a graph
In graph theory, a branch of mathematics, the (binary) cycle space of an undirected graph is the set of its even-degree spanning subgraphs, or the set
Cycle_space
Quantified formulas with real-number variables
complexity theory, it lies between NP and PSPACE. Many natural problems in geometric graph theory, especially problems of recognizing geometric intersection graphs
Existential theory of the reals
Existential_theory_of_the_reals
Algorithm for finding shortest paths
an algorithm for finding the shortest paths between nodes in a weighted graph, which may represent, for example, a road network. It was conceived by computer
Dijkstra's_algorithm
Length of shortest path between two nodes of a graph
mathematical field of graph theory, the distance between two vertices in a graph is the number of edges in a shortest path (also called a graph geodesic) connecting
Distance_(graph_theory)
Extremal graph theory bound on clique-free graph edges
In graph theory, Turán's theorem bounds the number of edges that can be included in an undirected graph that does not have a complete subgraph of a given
Turán's_theorem
Statement in mathematical combinatorics
(1984). "On some problems in graph theory, combinatorial analysis and combinatorial number theory" (PDF). Graph Theory and Combinatorics: 1–17. Kohayakawa
Ramsey's_theorem
Embedding a graph in a topological space, often Euclidean
In topological graph theory, an embedding (also spelled imbedding) of a graph G {\displaystyle G} on a surface Σ {\displaystyle \Sigma } is a representation
Graph_embedding
Visualization of node-link graphs
Graph drawing is an area of mathematics and computer science combining methods from geometric graph theory and information visualization to derive two-dimensional
Graph_drawing
Partial order with well-ordered predecessors
the sense of graph theory in one of two ways: either as a tree (graph theory) or as a trivially perfect graph. In the first case, the graph is the undirected
Tree_(set_theory)
Graph with at most one crossing per edge
In topological graph theory, a 1-planar graph is a graph that can be drawn in the Euclidean plane in such a way that each edge has at most one crossing
1-planar_graph
Branch of discrete mathematics
right. One of the oldest and most accessible parts of combinatorics is graph theory, which by itself has numerous natural connections to other areas. Combinatorics
Combinatorics
Two special graphs in graph theory
In the mathematical field of graph theory, the Klein graphs are two different but related regular graphs, each with 84 edges. Each can be embedded in
Klein_graphs
Size of biclique cover of a graph
fields of graph theory and combinatorial optimization, the bipartite dimension or biclique cover number of a graph G = (V, E) is the minimum number of bicliques
Bipartite_dimension
1979 conjecture in combinatorics
the intersection-closed formulation. Another equivalent formulation of the union-closed sets conjecture uses graph theory. In an undirected graph, an
Union-closed_sets_conjecture
16-regular graph with 27 vertices and 216 edges
the mathematical field of graph theory, the Schläfli graph, named after Ludwig Schläfli, is a 16-regular undirected graph with 27 vertices and 216 edges
Schläfli_graph
Graph drawn with all edges intersecting
A thrackle is an embedding of a graph in the plane in which each edge is a Jordan arc and every pair of edges meet exactly once. Edges may either meet
Thrackle
Subgraph with contracted edges
In graph theory, an undirected graph H is called a minor of the undirected graph G if H can be formed from G by deleting edges and vertices and by contracting
Graph_minor
Subset of a graph's nodes such that all other nodes link to at least one
In graph theory, a dominating set for a graph G is a subset D of its vertices, such that any vertex of G is in D, or has a neighbor in D. The domination
Dominating_set
Complexity class of problems
10th Ann. ACM Symp. on Theory of Computing. pp. 216–226. MR 0521057. Kisfaludi-Bak, Sándor (2020). "Hyperbolic intersection graphs and (quasi)-polynomial
NP-intermediate
Graph formed by touching unit circles
In geometric graph theory, a penny graph is a contact graph of unit circles. It is formed from a collection of unit circles that do not cross each other
Penny_graph
Trail in a graph that visits each edge once
In graph theory, an Eulerian trail (or Eulerian path) is a trail in a finite graph that visits every edge exactly once (allowing for revisiting vertices)
Eulerian_path
Distribution theory Dynamical systems theory Elimination theory Ergodic theory Extremal graph theory Field theory Galois theory Game theory Graph theory Group
List_of_mathematical_theories
extremal graph theory, the forbidden subgraph problem is the following problem: given a graph G {\displaystyle G} , find the maximal number of edges ex
Forbidden_subgraph_problem
Intersection graph representing regions on the Euclidean plane
In graph theory, a branch of mathematics, a map graph is an undirected graph formed as the intersection graph of finitely many simply connected and internally
Map_graph
degree theory Topological graph theory Topological K-theory Topos theory Toric geometry Transcendental number theory a branch of number theory that revolves
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Smallest dimension where a graph can be represented as an intersection graph of boxes
of graph theory, the boxicity of a graph is a graph invariant defined to be the minimum dimension of Euclidean space required to represent the graph as
Boxicity
American mathematician
mathematics, graph theory, and number theory. Hobbs and his colleague taught a course in the intersection of graph theory and number theory, he explains:
Arthur_Hobbs_(mathematician)
game Acyclic pebble game One-player pebble game Token on acyclic directed graph games: Quantified boolean formulas First-order logic of equality Provability
List of PSPACE-complete problems
List_of_PSPACE-complete_problems
In graph theory, a χ {\displaystyle \chi } -bounded (using the Greek letter chi) family F {\displaystyle {\mathcal {F}}} of graphs is one for which there
Chi-bounded
American mathematician
his contributions to graph theory, logic, and artificial intelligence. His research primarily focused on problems related to graph coloring, including
Landon_Rabern
Graph with at most one cycle per component
In graph theory, a pseudoforest is an undirected graph in which every connected component has at most one cycle. That is, it is a system of vertices and
Pseudoforest
On minimizing crossings in bicliques
bipartite graph be drawn with fewer crossings than the number given by Zarankiewicz? More unsolved problems in mathematics In the mathematics of graph drawing
Turán's_brick_factory_problem
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
Family of symmetric graphs which generalize the Petersen graph
of graph theory, the odd graphs are a family of symmetric graphs defined from certain set systems. They include and generalize the Petersen graph. The
Odd_graph
Conjecture about coloring graphs
vertex, then the union of the graphs can be properly colored with k colors. More unsolved problems in mathematics In graph theory, the Erdős–Faber–Lovász conjecture
Erdős–Faber–Lovász_conjecture
Computational problem of graph theory
In graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights
Shortest_path_problem
Intersection graph for a set of arcs on a circle
In graph theory, a circular-arc graph is the intersection graph of a set of arcs on the circle. It has one vertex for each arc in the set, and an edge
Circular-arc_graph
Notion in combinatorics
Sauer–Shelah lemma to prove results in graph theory such as that the number of strong orientations of a given graph is sandwiched between its numbers of
Sauer–Shelah_lemma
Mathematical problem
mathematics In geometric graph theory, the Hadwiger–Nelson problem, named after Hugo Hadwiger and Edward Nelson, asks for the minimum number of colors required
Hadwiger–Nelson_problem
Graph invariant defined from axis-parallel unit cubes
field of graph theory, cubicity is a graph invariant defined to be the smallest dimension such that a graph can be realized as the intersection graph of axis-parallel
Cubicity
Visual technique in topological graph theory
topological graph theory, a ribbon graph is a way to represent graph embeddings, equivalent in power to signed rotation systems and graph-encoded maps
Ribbon_graph
Supposition or system of ideas intended to explain something
theory — Galois theory — Game theory — Gauge theory — Graph theory — Group theory — Hodge theory — Homology theory — Homotopy theory — Ideal theory —
Theory
Mathematical result on infinite trees
theorem in graph theory due to the Hungarian mathematician Dénes Kőnig who published it in 1927. It gives a sufficient condition for an infinite graph to have
Kőnig's_lemma
Order dimension of incidences in planar graphs
In graph theory, Schnyder's theorem is a characterization of planar graphs in terms of the order dimension of their incidence posets. It is named after
Schnyder's_theorem
INTERSECTION NUMBER-GRAPH-THEORY
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INTERSECTION NUMBER-GRAPH-THEORY