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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
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
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 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
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
On bipartite matching and vertex cover
bipartite graph, can be interpreted as stating that the line graph of a bipartite graph is perfect. Since line graphs of bipartite graphs are perfect, the
Kőnig's theorem (graph theory)
Kőnig's_theorem_(graph_theory)
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
Graph where every connected induced subgraph has a universal vertex
In graph theory, a trivially perfect graph is a graph with the property that in each of its induced subgraphs the size of the maximum independent set equals
Trivially_perfect_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
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
Graph of chess rook moves
component of a decomposition of perfect graphs used to prove the strong perfect graph theorem, which characterizes all perfect graphs. The independence number
Rook's_graph
Graph where all long cycles have a chord
induced cycle in the graph should have exactly three vertices. The chordal graphs may also be characterized as the graphs that have perfect elimination orderings
Chordal_graph
Matching which covers every node of the graph
In graph theory, a perfect matching in a graph is a matching that covers every vertex of the graph. More formally, given a graph G with edges E and vertices
Perfect_matching
Partition of a graph into spanning subgraphs
of the graph into disjoint k-factors. A graph G is said to be k-factorable if it admits a k-factorization. In particular, a 1-factor is a perfect matching
Graph_factorization
Trail in which only the first and last vertices are equal
complement of a graph hole. Chordless cycles may be used to characterize perfect graphs: by the strong perfect graph theorem, a graph is perfect if and only
Cycle_(graph_theory)
Graph which partitions into a clique and independent set
Because chordal graphs are perfect, so are the split graphs. The double split graphs, a family of graphs derived from split graphs by doubling every
Split_graph
Topics referred to by the same term
"The Perfect", a song by the Killing Tree from The Romance of Helen Trent Perfect graph Perfect group Perfect lattice (same as perfect form) Perfect matrix
Perfect
Graph with same nodes as but complementary connections to another
the complement of a perfect graph is also perfect is the perfect graph theorem of László Lovász. Cographs are defined as the graphs that can be built up
Complement_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
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
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 whose maximal clique hypergraph is a hypertree
chordal graphs are exactly the strongly chordal graphs), and a dually chordal graph is in general not a perfect graph. Dually chordal graphs appeared
Dually_chordal_graph
Cycle graph plus universal vertex
In graph theory, a wheel graph is a graph formed by connecting a single universal vertex to all vertices of a cycle. A wheel graph with n vertices can
Wheel_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
Graph linking pairs of comparable elements in a partial order
Comparability graphs have also been called transitively orientable graphs, partially orderable graphs, containment graphs, and divisor graphs. An incomparability
Comparability_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 formed by adding isolated or universal vertices
split graph. Every graph that is both a trivially perfect graph and the complementary graph of a trivially perfect graph is a threshold graph. Threshold
Threshold_graph
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
On chains and antichains in partial orders
Every comparability graph is perfect: this is essentially just Mirsky's theorem, restated in graph-theoretic terms. By the perfect graph theorem of Lovász
Dilworth's_theorem
Graphs formed by a hypercube's edges and vertices
complete graph, and may be decomposed into two copies of Q n − 1 {\displaystyle Q_{n-1}} connected to each other by a perfect matching. Hypercube graphs should
Hypercube_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)
Special case of the perfect graphs in graph theory
the given graph. Perfectly orderable graphs form a special case of the perfect graphs, and they include the chordal graphs, comparability graphs, and distance-hereditary
Perfectly_orderable_graph
Property of graphs that depends only on abstract structure
In graph theory, a graph property or graph invariant is a property of graphs that depends only on the abstract structure, not on graph representations
Graph_property
Graph formed by complementation and disjoint union
special cases of the distance-hereditary graphs, permutation graphs, comparability graphs, and perfect graphs. Any cograph may be constructed using the
Cograph
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
Describing a family of graphs by excluding certain (sub)graphs
In graph theory, a branch of mathematics, many important families of graphs can be described by a finite set of individual graphs that do not belong to
Forbidden graph characterization
Forbidden_graph_characterization
Graph whose induced subgraphs preserve distance
graph is a perfect graph, more specifically a perfectly orderable graph and a Meyniel graph. Every distance-hereditary graph is also a parity graph,
Distance-hereditary_graph
complement of a disconnected graph. Skew partitions play an important role in the theory of perfect graphs. A skew partition of a graph G {\displaystyle G} is
Skew_partition
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
tolerance graphs themselves are perfect graphs. It is NP-complete to determine whether a given graph is a tolerance graph. However, because tolerance graphs are
Tolerance_graph
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
Graph path which is an induced subgraph
strong perfect graph theorem, the perfect graphs are the graphs with no odd hole and no odd antihole. The distance-hereditary graphs are the graphs in which
Induced_path
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
triangle-free graphs are bull-free graphs, since every bull contains a triangle. The strong perfect graph theorem was proven for bull-free graphs long before
Bull_graph
Characterizes the height of any finite partially ordered set
complement graph of a comparability graph is perfect. The perfect graph theorem of Lovász (1972) states that the complements of perfect graphs are always
Mirsky's_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
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
Graphs whose distances obey Ptolemy's inequality
graphs are exactly the graphs that are both chordal and distance-hereditary; they include the block graphs and are a subclass of the perfect graphs.
Ptolemaic_graph
In graph theory, an asteroidal triple-free graph or AT-free graph is a graph that contains no asteroidal triple. An asteroidal triple is an independent
Asteroidal_triple-free_graph
Graph often embedded in the Klein bottle
also a 3-vertex-connected and 3-edge-connected perfect graph. The automorphism group of the Franklin graph is of order 48 and is isomorphic to Z/2Z×S4,
Franklin_graph
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
Graph family made by joining complete graphs at a universal node
complete graphs from which it is formed, it is (k − 1)-edge-connected. It is trivially perfect and a block graph. By construction, the windmill graph Wd(3
Windmill_graph
Characterization of graphs with perfect matchings
discipline of graph theory, the Tutte theorem, named after William Thomas Tutte, is a characterization of finite undirected graphs with perfect matchings
Tutte's theorem on perfect matchings
Tutte's_theorem_on_perfect_matchings
Unsolved problem in computational complexity theory
computer science Can the graph isomorphism problem be solved in polynomial time? More unsolved problems in computer science The graph isomorphism problem is
Graph_isomorphism_problem
Mathematician and engineer
strong perfect graph theorem (with Neil Robertson, Paul Seymour, and Robin Thomas) characterizing perfect graphs as being exactly the graphs with no odd
Maria_Chudnovsky
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
British mathematician
theorem, linkless embeddings, graph minors and structure, the perfect graph conjecture, the Hadwiger conjecture, claw-free graphs, χ-boundedness, and the Erdős–Hajnal
Paul_Seymour_(mathematician)
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
One of two types of graph
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 of
Book_(graph_theory)
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
Subgraph induced by all nodes linked to a given node of a graph
graph in F is also locally F. For instance, every chordal graph is locally chordal; every perfect graph is locally perfect; every comparability graph
Neighbourhood_(graph_theory)
Graph made from a subset of another graph's nodes and their edges
In graph theory, an induced subgraph of a graph is another graph, formed from a subset of the vertices of the graph and all of the edges, from the original
Induced_subgraph
Conjecture in graph theory
spanning trees in a graph Chromatic polynomial Being a perfect graph or an interval graph, or certain other subclasses of perfect graphs Both the reconstruction
Reconstruction_conjecture
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
Indian mathematician
special case of the strong perfect graph conjecture for claw-free graphs. Parthasarathy guided and refereed Ph.D. students in graph theory, among them S. A
K. R. Parthasarathy (graph theorist)
K._R._Parthasarathy_(graph_theorist)
Award for advancements in discrete mathematics
that graph minors form a well-quasi-ordering. 2009: Maria Chudnovsky, Neil Robertson, Paul Seymour, and Robin Thomas, for the strong perfect graph theorem
Fulkerson_Prize
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
Undirected graph with 14 vertices
distance-transitive graph (see the Foster census) and therefore distance regular. There are 24 perfect matchings in the Heawood graph; for each matching
Heawood_graph
In graph theory and computer science, the graph sandwich problem is a problem of finding a graph that belongs to a particular family of graphs and is
Graph_sandwich_problem
Graph made from disjoint union of complete graphs
In graph theory, a branch of mathematics, a cluster graph is a graph formed from the disjoint union of complete graphs. Equivalently, a graph is a cluster
Cluster_graph
Mathematician and computer scientist (born 1948)
computer scientist known for his research on perfect graphs, graph sandwich problems, tolerance graphs, compiler optimization, and spatial-temporal reasoning
Martin_Charles_Golumbic
Partition of a graph's nodes into cliques
to the weak perfect graph theorem, the complement of a perfect graph is also perfect. Therefore, the perfect graphs are also the graphs in which, for
Clique_cover
French mathematician (1926–2002)
the Hungarian graph theorists. He published a survey paper on graph coloring, where he introduced the ideas that soon led to perfect graphs. In March 1960
Claude_Berge
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 in which every two vertices are adjacent
In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique
Complete_graph
Upper bound on a graph's Shannon capacity
graphs for which they are equal, including perfect graphs. Let G = ( V , E ) {\displaystyle G=(V,E)} be a graph on n {\displaystyle n} vertices. An ordered
Lovász_number
Class of mathematical graphs
bisplit graph is a comparability graph, and hence also a perfect graph. This can be shown by constructing a transitive orientation of these graphs, an assignment
Bisplit_graph
Graph in graph theory
In graph theory, the lexicographic product or (graph) composition G ∙ H of graphs G and H is a graph such that the vertex set of G ∙ H is the cartesian
Lexicographic product of graphs
Lexicographic_product_of_graphs
Moroccan mathematician
strong perfect graph theorem, for the graphs that have no bull graph as an induced subgraph.[B] Their work in this area introduced a type of graph decomposition
Najiba_Sbihi
Graph of n vertices with a perfect matching for every subgraph of n-1 vertices
results in a graph with a perfect matching, a way of grouping the remaining vertices into adjacent pairs. A matching of all but one vertex of a graph is called
Factor-critical_graph
Algorithm for counting perfect matchings in planar graphs
counts the number of perfect matchings in a planar graph in polynomial time. This same task is #P-complete for general graphs. For matchings that are
FKT_algorithm
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
Conjecture in graph theory
conjecture is that for every graph H {\displaystyle H} , the H {\displaystyle H} -free graphs necessarily contain a perfect induced subgraph of polynomial
Erdős–Hajnal_conjecture
Mathematical graph theorem
Petersen's Theorem. Every cubic, bridgeless graph contains a perfect matching. In other words, if a graph has exactly three edges at each vertex, and
Petersen's_theorem
Chordal graph where all cycles of even length have odd chords
u1w1u2w2... has no odd chord. Strongly chordal graphs may also be characterized as the graphs having a strong perfect elimination ordering, an ordering of the
Strongly_chordal_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
Result in combinatorics and graph theory
number of sets in the subset. The graph theoretic formulation answers whether a finite bipartite graph has a perfect matching—that is, a way to match each
Hall's_marriage_theorem
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
American mathematician (born 1950)
include facility location, integer programming, balanced matrices, and perfect graphs. Cornuéjols graduated from École nationale des ponts et chaussées and
Gérard_Cornuéjols
Zverovich (2000) defines a class of perfect cochromatic graphs, analogous to the definition of perfect graphs via graph coloring, and provides a forbidden
Cocoloring
Mathematician (1962–2020)
on the Hadwiger conjecture, and in 2009 for the proof of the strong perfect graph theorem. In 2011, he was awarded the Karel Janeček Foundation Neuron
Robin_Thomas_(mathematician)
Bipartite graph partition with special property
in a perfect matching of the graph. It is named after A. L. Dulmage and Nathan Mendelsohn, who published it in 1958. A generalization to any graph is the
Dulmage–Mendelsohn decomposition
Dulmage–Mendelsohn_decomposition
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
Canadian-American mathematician (born 1938)
4-colorings of planar graphs. In 2006, Robertson, Seymour, Thomas, and Maria Chudnovsky, proved the long-conjectured strong perfect graph theorem characterizing
Neil Robertson (mathematician)
Neil_Robertson_(mathematician)
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)
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
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
Task of computing complete subgraphs
to other, non-perfect, classes of graphs as well. For instance, in a circle graph, the neighborhood of each vertex is a permutation graph, so a maximum
Clique_problem
each direction. When a graph has a Pfaffian orientation, the orientation can be used to count the perfect matchings of the graph. This is the main idea
Pfaffian_orientation
Recursively splitting a graph into subsets of nodes
distance-hereditary graphs (Spinrad, 2003) and for graph drawing (Papadopoulos, 2006). They play an important role in Lovász's celebrated proof of the perfect graph theorem
Modular_decomposition
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