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Graph with tight clique-coloring relation
In graph theory, a perfect graph is a graph in which the chromatic number equals the size of the maximum clique, both in the graph itself and in every
Perfect_graph
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
Graph whose line graph is perfect
In graph theory, a line perfect graph is a graph whose line graph is a perfect graph. Equivalently, these are the graphs in which every odd-length simple
Line_perfect_graph
Perfect graphs have neither odd holes nor odd antiholes
In graph theory, the strong perfect graph theorem is a forbidden graph characterization of the perfect graphs as being exactly the graphs that have neither
Strong_perfect_graph_theorem
Complements of perfect graphs are perfect
In graph theory, the perfect graph theorem of László Lovász (1972a, 1972b) states that an undirected graph is perfect if and only if its complement graph
Perfect_graph_theorem
Graph of chess rook moves
mathematics of graphs through their alternative constructions: rook's graphs are the Cartesian product of two complete graphs, and are the line graphs of complete
Rook's_graph
independence number of its line graph. Similarly, χ(G) is the chromatic number of a graph; χ ′(G) is the chromatic index of the graph, which equals the chromatic
Glossary_of_graph_theory
Intersection graph for intervals on the real number line
intersection graph of the intervals. Interval graphs are chordal graphs and perfect graphs. They can be recognized in linear time, and an optimal graph coloring
Interval_graph
On bipartite matching and vertex cover
the line graph of a bipartite graph is perfect. Since line graphs of bipartite graphs are perfect, the complements of line graphs of bipartite graphs are
Kőnig's theorem (graph theory)
Kőnig's_theorem_(graph_theory)
One of two types of graph
key building blocks of line perfect graphs. The term "book-graph" has been employed for other uses. Barioli used it to mean a graph composed of a number
Book_(graph_theory)
Graph without four-vertex star subgraphs
characterization of claw-free perfect graphs. They are the subject of hundreds of mathematical research papers and several surveys. The line graph L ( G ) {\displaystyle
Claw-free_graph
Trail in which only the first and last vertices are equal
directed graph with no directed cycles Forest, a cycle-free graph Line perfect graph, a graph in which every odd cycle is a triangle Perfect graph, a graph with
Cycle_(graph_theory)
Graph divided into two independent sets
results concerns perfect graphs: every bipartite graph, the complement of every bipartite graph, the line graph of every bipartite graph, and the complement
Bipartite_graph
Set of edges without common vertices
three graphs. A perfect matching is a matching that matches all vertices of the graph. That is, a matching is perfect if every vertex of the graph is incident
Matching_(graph_theory)
Graph where all odd cycles of length ≥ 5 has 2+ chords
Meyniel graphs are a subclass of the perfect graphs. Every induced subgraph of a Meyniel graph is another Meyniel graph, and in every Meyniel graph the size
Meyniel_graph
Describing a family of graphs by excluding certain (sub)graphs
perfect graphs", Discrete Mathematics, 24 (1): 105–107, doi:10.1016/0012-365X(78)90178-4. Metelsky, Yury; Tyshkevich, Regina (1997), "On line graphs of
Forbidden graph characterization
Forbidden_graph_characterization
Cubic graph with 10 vertices and 15 edges
bridgeless graph has a cycle-continuous mapping to the Petersen graph. More unsolved problems in mathematics In the mathematical field of graph theory, the
Petersen_graph
Graph where any two induced paths between nodes both have odd or even lengths
same parity, and the line perfect graphs, a generalization of the bipartite graphs. Every parity graph is a Meyniel graph, a graph in which every odd cycle
Parity_graph
Graph representing a permutation
reversed by the permutation. Permutation graphs may also be defined geometrically, as the intersection graphs of line segments whose endpoints lie on two parallel
Permutation_graph
Methodic assignment of colors to elements of a graph
coloring of a graph is just a vertex coloring of its line graph, and a face coloring of a plane graph is just a vertex coloring of its dual. However, non-vertex
Graph_coloring
Graph whose biconnected components are all cliques
subclasses of the perfect graphs, block graphs are perfect. Every tree, cluster graph, or windmill graph is a block graph. Every block graph has boxicity at
Block_graph
Undirected cubic graph with 12 vertices and 18 edges
In the mathematical field of graph theory, Tietze's graph is an undirected cubic graph with 12 vertices and 18 edges. It is named after Heinrich Franz
Tietze's_graph
Intersection graph of trapezoids between parallel lines
points on the bottom line. A graph is a trapezoid graph if there exists a set of trapezoids corresponding to the vertices of the graph such that two vertices
Trapezoid_graph
Concept in graph theory
In graph theory, a strongly regular graph (SRG) is a regular graph G = (V, E) with v vertices and degree k such that for some given integers λ , μ ≥ 0
Strongly_regular_graph
Graph where every edge is in one triangle
vertex) the rest of the graph looks like a perfect matching. Locally linear graphs have also been called locally matched graphs. More technically, the
Locally_linear_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 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
Adjacent subset of an undirected graph
cover. A perfect graph is a graph in which the clique number equals the chromatic number in every induced subgraph. A split graph is a graph in which
Clique_(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
Graph of intervisible locations in computational geometry
These graphs do not fall into many known families of well-structured graphs: they might not be perfect graphs, circle graphs, or chordal graphs. An exception
Visibility_graph
Graph with all vertices of degree 4
mathematical field of graph theory, a quartic graph is a graph where all vertices have degree 4. In other words, a quartic graph is a 4-regular graph. Several well-known
Quartic_graph
Cocomparability graph Perfect graph Lekkerkerker, C. G.; Boland, J. Ch. (1962), "Representation of a finite graph by a set of intervals on the real line", Fundamenta
Asteroidal_triple-free_graph
Vertices connected in pairs by edges
graph is a forest. More advanced kinds of graphs are: Petersen graph and its generalizations; perfect graphs; cographs; chordal graphs; other graphs with
Graph_(discrete_mathematics)
k} -regular graph with 2 n {\displaystyle 2n} vertices is 1-factorable. The perfect 1-factorization conjecture that every complete graph on an even number
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Unsolved problem in computational complexity theory
bipartite Eulerian graphs bipartite regular graphs line graphs split graphs chordal graphs regular self-complementary graphs polytopal graphs of general, simple
Graph_isomorphism_problem
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
Intersection graph of unit intervals on the real line
interval graph; in particular, it is a special case of a chordal graph and of a perfect graph. It is also a special case of a circle graph, something
Indifference_graph
Maximum number of colors in a greedy graph coloring
least three. The crown graphs are obtained from complete bipartite graphs K n , n {\displaystyle K_{n,n}} by removing a perfect matching. As a result,
Grundy_number
Theorem in combinatorics
bipartite multigraphs, and more generally to line-perfect graphs by way of Maffray's characterization of the line graphs possessing kernels. Alexandr Kostochka
Dinitz_theorem
The line graph of a regular integral graph is again integral. For instance, as the line graph of K 4 {\displaystyle K_{4}} , the octahedral graph is integral
Integral_graph
Graph formed by subdivision of triangles
planar 3-trees, the maximal planar chordal graphs, the uniquely 4-colorable planar graphs, and the graphs of stacked polytopes. They are named after Apollonius
Apollonian_network
Graph whose peripheral cycles are all triangles
graphs are exactly the graphs that can be formed as clique-sums of complete graphs and maximal planar graphs. Line perfect graph, a graph in which every odd
Strangulated_graph
Two-player paper-and-pencil game
three colors to paint an edge of the graph, until a player loses by completing a monochromatic triangle. Finding perfect winning strategies for these variants
Sim_(game)
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
Intersection graph for a set of arcs on a circle
stretched to a line, which results in an interval representation. Unlike interval graphs, however, circular-arc graphs are not always perfect, as the odd
Circular-arc_graph
Graph showing unequal income growth
elephant curve, also known as the Lakner-Milanovic graph or the global growth incidence curve, is a graph that illustrates the unequal distribution of income
Elephant_curve
Argentine-born American mathematician
two copies of the Ljubljana graph. See also. Moreover, relations of this subject with square-blocking subsets and with perfect dominating sets (see below)
Italo_Jose_Dejter
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)
Sharpest angle between edges at a vertex
the drawing. Formann et al. (1993) observed that every straight-line drawing of a graph with maximum degree d has angular resolution at most 2π/d: if v
Angular resolution (graph drawing)
Angular_resolution_(graph_drawing)
Three raised to an integer power
regular graphs also have a number of vertices that is a power of three, including the Brouwer–Haemers graph (81 vertices), Berlekamp–van Lint–Seidel graph (243
Power_of_three
In graph theory, an independent dominating set for a graph G = ( V , E ) {\displaystyle G=(V,E)} is a subset D ⊆ V {\displaystyle D\subseteq V} that is
Independent_dominating_set
chordal graphs, because the squares of line graphs of chordal graphs are perfect graphs. Moreover, it can be solved in linear time in chordal graphs . Unless
Induced_matching
Graph whose vertices correspond to combinations of a set of n elements
Kneser graph K(n, 2) is the complement of the line graph of the complete graph on n vertices. The Kneser graph K(2n − 1, n − 1) is the odd graph On; in
Kneser_graph
Decomposition of a graph into hamiltonion cycles
In graph theory, a branch of mathematics, a Hamiltonian decomposition of a given graph is a partition of the edges of the graph into Hamiltonian cycles
Hamiltonian_decomposition
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
Natural number
The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31. "Sloane's A082897 : Perfect totient numbers". The On-Line Encyclopedia
39_(number)
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
Vertex adjacent to all others in a graph
logic of graphs, and for apex graphs. Graphs that contain a universal vertex include the stars, trivially perfect graphs, and friendship graphs. For wheel
Universal_vertex
Undirected graph with 11 nodes and 27 edges
In the mathematical field of graph theory, the Goldner–Harary graph is a simple undirected graph with 11 vertices and 27 edges. It is named after Anita
Goldner–Harary_graph
Family of graphs based on the Fibonacci sequence
resonance graph or (Z-transformation graph) of G is a graph whose vertices describe perfect matchings of G and whose edges connect pairs of perfect matchings
Fibonacci_cube
Balanced complete multipartite graph
in a Turán graph lead to notable graphs that have been independently studied. The Turán graph T(2n,n) can be formed by removing a perfect matching from
Turán_graph
Assignment of colors to edges of a graph
In graph theory, a proper edge coloring of a graph is an assignment of "colors" to the edges of the graph so that no two incident edges have the same color
Edge_coloring
Graph with equal-size maximal independent sets
In graph theory, a well-covered graph is an undirected graph in which the minimal vertex covers all have the same size. Here, a vertex cover is a set
Well-covered_graph
Graph generated by a random process
In mathematics, random graph is the general term to refer to probability distributions over graphs. Random graphs may be described simply by a probability
Random_graph
Representation of a graph as a path graph "thickened" by some amount
In graph theory, a path decomposition of a graph G is, informally, a representation of G as a "thickened" path graph, and the pathwidth of G is a number
Pathwidth
Contour line in microeconomics
right of the contour line unless they exceed their constraints. A family of isoquants can be represented by an isoquant map, a graph combining a number
Isoquant
One-by-one assignment of colors to graph vertices
\beta } -perfect graphs. If a graph and its complement graph are both even-hole-free, they are both β {\displaystyle \beta } -perfect. The graphs that are
Greedy_coloring
Automated method for solving mazes
"simply connected", or "perfect" mazes, and are equivalent to a tree in graph theory. Maze-solving algorithms are closely related to graph theory. Intuitively
Maze-solving_algorithm
Polyhedral graph with 26 vertices and 39 edges
In the mathematical field of graph theory, the 26-fullerene graph is a polyhedral graph with V = 26 vertices and E = 39 edges. Its planar embedding has
26-fullerene_graph
Measurement of graph sparsity
In graph theory, a k-degenerate graph is an undirected graph in which every non-empty subgraph has at least one vertex of degree at most k {\displaystyle
Degeneracy_(graph_theory)
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
Natural number
"Perfect Number". mathworld.wolfram.com. Retrieved 2025-03-20. Sloane, N. J. A. (ed.). "Sequence A002827 (Unitary perfect numbers)". The On-Line Encyclopedia
6
Assignment problem in combinatorial mathematics
be interpreted in graph-theoretic terms, as directed Hamiltonian cycles in crown graphs. A crown graph is formed by removing a perfect matching from a complete
Ménage_problem
Natural number
On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed.). "Sequence A000664 (Number of graphs with n edges)". The On-Line Encyclopedia
177_(number)
Graph drawing with vertices on a line
An arc diagram is a style of graph drawing, in which the vertices of a graph are placed along a line in the Euclidean plane and edges are drawn using
Arc_diagram
Python library for graphs and networks
NetworkX is a Python library for studying graphs and networks. NetworkX is free software released under the BSD-new license. NetworkX began development
NetworkX
Natural number
The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2 June 2022. Hougardy, Stefan (October 2006). "Classes of perfect graphs". Discrete
700_(number)
Natural number
less than n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Zachariou, Andreas; Zachariou, Eleni (1972). "Perfect, Semiperfect and Ore
54_(number)
Fewest graph edges whose removal breaks all cycles
In graph theory, a branch of mathematics, the cyclomatic number, circuit rank, cycle rank, corank or nullity of an undirected graph is the minimum number
Cyclomatic_number
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
Generalizations in graph theory
theorem provides a condition guaranteeing that a bipartite graph (X + Y, E) admits a perfect matching, or - more generally - a matching that saturates
Hall-type theorems for hypergraphs
Hall-type_theorems_for_hypergraphs
Limited form of tree data structure
and S is a singleton (a single–element set) containing the root. From a graph theory perspective, binary trees as defined here are arborescences. A binary
Binary_tree
Natural number
labeled)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed.). "Sequence A002494 (Number of n-node graphs without isolated
888_(number)
Algebraic encoding of graph connectivity
is a graph polynomial. It is a polynomial in two variables which plays an important role in graph theory. It is defined for every undirected graph G {\displaystyle
Tutte_polynomial
Least-weight tree connecting graph vertices
graph theory, a minimum spanning tree (MST) or minimum weight spanning tree is a subset of the edges of a connected, edge-weighted undirected graph that
Minimum_spanning_tree
Word processing application
spreadsheet capabilities and the possibility to generate graphs) are also notable. The WordPerfect document format allows continuous extending of functionality
WordPerfect
Combinitorics of Polyhedra
its vertices can be thought of as describing all perfect matchings in a complete bipartite graph, and a linear optimization problem on this polytope
Polyhedral_combinatorics
Measure of graph complexity
In graph theory, the clique-width of a graph G is a parameter that describes the structural complexity of the graph; it is closely related to treewidth
Clique-width
Order by which an agent ranks alternatives based on their utility
straight line is lower than that of those two points, this is a strictly concave preference. Straight-line similarities occur when there are perfect substitutes
Preference_(economics)
Graph of wealth or income distribution
for representing inequality of the wealth distribution. The curve is a graph showing the proportion of overall income or wealth assumed by the bottom
Lorenz_curve
Natural number
N. J. A. (ed.). "Sequence A052431 (Number of perfect simple undirected graphs on n nodes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation
148_(number)
Solid with six equal square faces
drawing a graph with vertices connected with an edge in a plane. Such a graph is called the cubical graph, a special case of the hypercube graph. The cube
Cube
algorithm: convert a bipartite graph to a maximum-cardinality matching Hungarian algorithm: algorithm for finding a perfect matching Prüfer coding: conversion
List_of_algorithms
-bounded graph families include: The perfect graphs, with f ( t ) = t {\displaystyle f(t)=t} The graphs of boxicity two, which is the intersection graphs of
Chi-bounded
11th Johnson solid (16 faces)
the icosahedral graph's vertices, leaving 11 vertices, an odd number, resulting in a graph with a perfect matching. Hence, the graph is a 2-vertex connected
Gyroelongated pentagonal pyramid
Gyroelongated_pentagonal_pyramid
Automated methods for the creation of mazes
between them. This predetermined arrangement can be considered as a connected graph with the edges representing possible wall sites and the nodes representing
Maze_generation_algorithm
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
Network whose degree distribution follows a power law
transformation which converts random graphs to their edge-dual graphs (or line graphs) produces an ensemble of graphs with nearly the same degree distribution
Scale-free_network
German mathematician (1871–1928)
thesis also contains the proof of Kőnig's theorem for regular bipartite graphs, phrased in the language of configurations. In 1910 Steinitz published the
Ernst_Steinitz
the edge dominating set problem, i.e., the dominating set problem in line graphs. NP-complete variants include the connected dominating set problem and
List_of_NP-complete_problems
travel, tourism, insurance
LINE PERFECT-GRAPH
LINE PERFECT-GRAPH
LINE PERFECT-GRAPH
LINE PERFECT-GRAPH
LINE PERFECT-GRAPH
LINE PERFECT-GRAPH
LINE PERFECT-GRAPH
LINE PERFECT-GRAPH
LINE PERFECT-GRAPH
travel, tourism, insurance