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Graph linking pairs of comparable elements in a partial order
In graph theory and order theory, a comparability graph is an undirected graph that connects pairs of elements that are comparable to each other in a
Comparability_graph
On chains and antichains in partial orders
any two comparable elements. Thus, a clique in a comparability graph corresponds to a chain, and an independent set in a comparability graph corresponds
Dilworth's_theorem
Property of elements related by inequalities
Look up comparability in Wiktionary, the free dictionary. In mathematics, two elements x and y of a set P are said to be comparable with respect to a
Comparability
Graph with tight clique-coloring relation
graph. Finite comparability graphs (and their complementary incomparability graphs) are always perfect. A clique, in a comparability graph, comes from a
Perfect_graph
Intersection graph for intervals on the real number line
complement is a comparability graph, it follows that graph and its complement are both interval graphs if and only if the graph is both a split graph and a permutation
Interval_graph
are comparable in the partial order. Equivalently, a comparability graph is a graph that has a transitive orientation. Many other classes of graphs can
Glossary_of_graph_theory
Recursively splitting a graph into subsets of nodes
graphs) and is useful to design efficient algorithms for the recognition of some graph classes, for finding transitive orientations of comparability graphs
Modular_decomposition
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)
relationship in directed trees and directed series–parallel graphs. The comparability graphs of series-parallel partial orders are cographs. Series-parallel
Series-parallel_partial_order
Graph where every connected induced subgraph has a universal vertex
that such a graph is perfect." Trivially perfect graphs are also known as comparability graphs of trees, arborescent comparability graphs, and quasi-threshold
Trivially_perfect_graph
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
Partially ordered set equipped with a rank function
connected component of its comparability graph is graded, so further characterizations will suppose this comparability graph to be connected. On each connected
Graded_poset
Graph which partitions into a clique and independent set
split graph and an interval graph, then its complement is both a split graph and a comparability graph, and vice versa. The split comparability graphs, and
Split_graph
Characterizes the height of any finite partially ordered set
induced subgraphs of comparability graphs are themselves comparability graphs, so Mirsky's theorem states that comparability graphs are perfect. Analogously
Mirsky's_theorem
Graph representing a permutation
{\overline {G}}} are comparability graphs. A graph G {\displaystyle G} is a permutation graph if and only if it is the comparability graph of a partially ordered
Permutation_graph
important in graph theory, since they generalise several important classes of graphs, e.g. circle graphs, 3-colorable graphs and comparability graphs. It was
Word-representable_graph
Well-quasi-ordering of finite trees
transfinite recursion). In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has also proved important
Kruskal's_tree_theorem
Graph representing intersections between given sets
permutation graph, in turn they are a special case of the family of the complements of comparability graphs known as cocomparability graphs. A unit disk graph is
Intersection_graph
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
Assigning directions to the edges of an undirected graph
the resulting directed graph is its own transitive closure. The graphs with transitive orientations are called comparability graphs; they may be defined
Orientation_(graph_theory)
Algorithmically defined graph
representing a graph as a unit disk graph may require exponentially many bits for the coordinates of the disk centers. Low-dimensional comparability graphs The comparability
Implicit_graph
Mathematical set with an ordering
set, a poset-based approach to quantum gravity Comparability graph – Graph linking pairs of comparable elements in a partial order Complete partial order –
Partially_ordered_set
Problem of finding the longest simple path for a given graph
circular-arc graphs and of co-comparability graphs (i.e. of the complements of comparability graphs, which also contain permutation graphs), both having
Longest_path_problem
Complements of perfect graphs are perfect
Perfect graphs include many important graphs classes including bipartite graphs, chordal graphs, and comparability graphs. The complement of a graph has an
Perfect_graph_theorem
Class of artificial neural networks
Graph neural networks (GNNs) are artificial neural networks designed for tasks whose inputs are graphs. Because graphs usually do not have a canonical
Graph_neural_network
Database using graph structures for queries
A graph database (GDB) is a database that uses graph structures for semantic queries with nodes, edges, and properties to represent and store data. A key
Graph_database
Smallest complete lattice containing a partial order
maximal independent set in the comparability graph of Q, or a maximal clique in the complement of the comparability graph, so algorithms for the clique
Dedekind–MacNeille_completion
Mathematical measure for partial orders
comparability graphs of the partial orders of dimension two are exactly the permutation graphs, graphs that are both themselves comparability graphs and
Order_dimension
Intersection graph for curves in the plane
complement graph of every comparability graph (also known as the cocomparability graph) is also a string graph. Kratochvíl (1991b) showed string graph recognition
String_graph
Representation of a graph as a path graph "thickened" by some amount
time algorithm for comparability graphs of interval orders generalizes this result, since any chordal graph must be a comparability graph of this type. Suchan
Pathwidth
then the corresponding comparability graph is bipartite. If the ordering is a total order, then the corresponding comparability graph is complete. If neither
Skew_partition
Order-preserving mathematical function
The graph of a monotone operator G ( T ) {\displaystyle G(T)} is a monotone set. A monotone operator is said to be maximal monotone if its graph is a
Monotonic_function
Perfect graphs have neither odd holes nor odd antiholes
graph; perfect graphs include many well-known graph classes including the bipartite graphs, chordal graphs, and comparability graphs. In his 1961 and
Strong_perfect_graph_theorem
Property of a relation on a set
meaning, which applies to precisely those orders whose comparability graphs are connected graphs. This applies, for instance, to fences, of which none
Connected_relation
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
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
Visual depiction of a partially ordered set
automatically using graph drawing techniques. In some sources, the phrase "Hasse diagram" has a different meaning: the directed acyclic graph obtained from
Hasse_diagram
Order whose elements are all comparable
polynomials. Another example is the use of "chain" as a synonym for a walk in a graph. One may define a totally ordered set as a particular kind of lattice, namely
Total_order
Generalization of graph theory
hypergraph is a generalization of a graph in which an edge can join any number of vertices. In contrast, in an ordinary graph, an edge connects exactly two
Hypergraph
Generalised alphabetical order
Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism Embedding
Lexicographic_order
Abstract simplicial complex describing a graph's cliques
interpreted as the clique complex of the comparability graph of the partial order. The matching complex of a graph consists of the sets of edges no two of
Clique_complex
Type of monotone function
substructure of B. (Graph theoretically) A poset is a (transitive, acyclic, directed, reflexive) graph. An order embedding A → B is a graph isomorphism from
Order_embedding
Mathematical ranking of a set
instantiation, and that is assumed to implement a strict weak ordering. Comparability – Property of elements related by inequalities Preorder – Reflexive
Weak_ordering
Matching which covers every node of the graph
Umesh V.; Vazirani, Vijay V. (1985). "NC algorithms for comparability graphs, interval graphs, and testing for unique perfect matching". In Maheshwari
Perfect_matching
Relationship between elements of two sets
relations leans on graph theory: For relations on a set (homogeneous relations), a directed graph illustrates a relation and a graph a symmetric relation
Binary_relation
Certain topology in mathematics
Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism Embedding
Order_topology
Branch of mathematics
giving directions to the edges of a graph. In this way, each order is seen to be equivalent to a directed acyclic graph, where the nodes are the elements
Order_theory
Smallest transitive relation containing a given binary relation
closure and transitive reduction are also used in the closely related area of graph theory. A relation R on a set X is transitive if, for all x, y, z in X,
Transitive_closure
Alternative mathematical ordering
group. Another formulation is to make X into the standard directed cycle graph on n vertices, by some matching of elements to vertices. It can be instinctive
Cyclic_order
empty. The graph sandwich problem is NP-complete when Π is the property of being a chordal graph, comparability graph, permutation graph, chordal bipartite
Graph_sandwich_problem
Mathematical operation
operation sequence. The small circle was used in the introductory pages of Graphs and Relations until it was dropped in favor of juxtaposition (no infix notation)
Composition_of_relations
Undirected, connected, and acyclic graph
In graph theory, a tree is an undirected graph in which every pair of distinct vertices is connected by exactly one path, or equivalently, a connected
Tree_(graph_theory)
Mathematical property of subsets in order theory
Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism Embedding
Cofinal_(mathematics)
Canadian graph theorist
algorithms in graph theory and special classes of graphs, including cographs, permutation graphs, interval graphs, comparability graphs and their complements
Lorna_Stewart
semiorders. The complement of the comparability graph of an interval order ( X {\displaystyle X} , ≤) is the interval graph ( X , ∩ ) {\displaystyle (X,\cap
Interval_order
Nonempty, upper-bounded, downward-closed subset
Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism Embedding
Ideal_(order_theory)
Glossary of terms used in branch of mathematics
finite. Comparable. Two elements x and y of a poset P are comparable if either x ≤ y or y ≤ x. Comparability graph. The comparability graph of a poset
Glossary_of_order_theory
Reversal of the order of elements of a binary relation
f^{-1}\subseteq Y\times X} defined by the graph f − 1 = { ( y , x ) ∈ Y × X : y = f ( x ) } . {\displaystyle \operatorname {graph} \,f^{-1}=\{(y,x)\in Y\times X:y=f(x)\}
Converse_relation
Conjunctive normal form Disjunctive normal form Formal system And-inverter graph Logic gate Boolean analysis Boolean prime ideal theorem Compactness theorem
List of Boolean algebra topics
List_of_Boolean_algebra_topics
Task of computing complete subgraphs
permutation graph. Even, Pnueli & Lempel (1972) provide an alternative quadratic-time algorithm for maximum cliques in comparability graphs, a broader
Clique_problem
Construction in order theory
Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism Embedding
Product_order
Type of graph in mathematics
Jung, H. A. (1978), "On a class of posets and the corresponding comparability graphs", Journal of Combinatorial Theory, Series B, 24 (2): 125–133, doi:10
Multitree
Special subset of a partially ordered set
Igarashi, Ayumi; Zwicker, William S. (16 February 2021). "Fair division of graphs and of tangled cakes". arXiv:2102.08560 [math.CO]. Davey, B. A.; Priestley
Filter_(mathematics)
Binary relation over a set and itself
endorelations. Terminology particular for graph theory is used for description, with an ordinary (undirected) graph presumed to correspond to a symmetric
Homogeneous_relation
Mathematical result on order relations
pairs incomparable can be extended in such a way that every pair becomes comparable. The theorem is one of many examples of the use of the axiom of choice
Szpilrajn_extension_theorem
Bound lattice in which every element has a complement
Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism Embedding
Complemented_lattice
Special type of lattice
elements. If a lattice is distributive, its covering relation forms a median graph. Furthermore, every distributive lattice is also modular. The introduction
Distributive_lattice
Mathematical concept for comparing objects
Ketonen. Finite graphs ordered by a notion of embedding called "graph minor" is a well-quasi-order (Robertson–Seymour theorem). Graphs of finite tree-depth
Well-quasi-ordering
Reflexive and transitive binary relation
graph. A preorder that is symmetric is an equivalence relation; it can be thought of as having lost the direction markers on the edges of the graph.
Preorder
Element of graph theory
every graph has a transitive orientation; the graphs that do are the comparability graphs. Complete graphs are special cases of comparability graphs, and
Acyclic_orientation
Vertex adjacent to all others in a graph
1090/gsm/089, ISBN 978-0-8218-4467-0, MR 2389013. Wolk, E. S. (1962), "The comparability graph of a tree", Proceedings of the American Mathematical Society, 13
Universal_vertex
Algebraic structure modeling logical operations
Free Boolean algebra Heyting algebra Hypercube graph Karnaugh map Laws of Form Logic gate Logical graph Logical matrix Propositional logic Quine–McCluskey
Boolean_algebra_(structure)
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
Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism Embedding
Symmetric_closure
Mathematical proposition equivalent to the axiom of choice
not every pair of elements of a partially ordered set is required to be comparable under the order relation, that is, in a partially ordered set P with order
Zorn's_lemma
2003), comparability graph with maximum degree 4 (Ochem 2017), line graph of a bipartite graph with maximum degree 4 (Gonçalves & Ochem 2009), graph with
Subcoloring
Construction in order theory
Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism Embedding
Star_product
Set whose pairs have minima and maxima
Bloom filter Information flow Ordinal optimization Quantum logic Median graph Knowledge space Regular language learning Analogical modeling Grätzer 2003
Lattice_(order)
Set theory concept
Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism Embedding
Club_set
Algebraic object with an ordered structure
Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism Embedding
Ordered_field
Type of topology in mathematics
Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism Embedding
Alexandrov_topology
Smallest dimension where a graph can be represented as an intersection graph of boxes
comparability graph of P (which is bipartite, since P has height two). Many graph problems can be solved or approximated more efficiently for graphs with
Boxicity
Graph in climate science
Hockey stick graphs present global or hemispherical mean global surface temperatures of the distant past, as shown by quantitative climate reconstructions
Hockey stick graph (global temperature)
Hockey_stick_graph_(global_temperature)
Size of subsets in order theory
Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism Embedding
Cofinality
Type of binary relation
relation (N, S), where N is the set of all natural numbers, and S is the graph of the successor function x ↦ x+1. Then induction on S is the usual mathematical
Well-founded_relation
Type of logical relation
University Schmidt, Gunther; Ströhlein, Thomas (6 December 2012). Relations and Graphs: Discrete Mathematics for Computer Scientists. Springer Science & Business
Total_relation
Equivalence of partially ordered sets
Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism Embedding
Order_isomorphism
There are equally many countable order types and real numbers
Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism Embedding
Cantor–Bernstein_theorem
Concept in order theory
Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism Embedding
Join_and_meet
Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism Embedding
Eulerian_poset
Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism Embedding
Absolutely and completely monotonic functions and sequences
Absolutely_and_completely_monotonic_functions_and_sequences
Banach space with a compatible structure of a lattice
Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism Embedding
Banach_lattice
Numerical ordering with a margin of error
Theorem 3 describes a more general situation in which the threshold for comparability between two utilities is a function of the utility rather than being
Semiorder
Ideals in a Boolean algebra can be extended to prime ideals
axiom of choice. In graph theory, the de Bruijn–Erdős theorem is another equivalent to BPI. It states that, if a given infinite graph requires at least
Boolean_prime_ideal_theorem
Isomorphism type of ordered sets
Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism Embedding
Order_type
Subset of a preorder that contains all larger elements
Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism Embedding
Upper_and_lower_sets
Mathematical ordering with upper bounds
to its right, then a {\displaystyle a} and b {\displaystyle b} are not comparable, and the subset { a , b } {\displaystyle \{a,b\}} has no upper bound)
Directed_set
Subset of incomparable elements
"The complexity of counting cuts and of computing the probability that a graph is connected", SIAM Journal on Computing, 12 (4): 777–788, doi:10.1137/0212053
Antichain
Partially ordered topological space
closed partial order ≤ {\displaystyle \leq } , i.e. a partial order whose graph { ( x , y ) ∈ X 2 ∣ x ≤ y } {\displaystyle \{(x,y)\in X^{2}\mid x\leq y\}}
Partially_ordered_space
Class of mathematical orderings
Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism Embedding
Well-order
COMPARABILITY GRAPH
COMPARABILITY GRAPH
Boy/Male
Spanish American Italian Latin
Enduring. The poet Dante Alighieri wrote The Divine Comedy with its graphic description of...
Boy/Male
Italian Spanish
Enduring. The poet Dante Alighieri wrote The Divine Comedy with its graphic description of...
Boy/Male
Italian Spanish
Enduring. The poet Dante Alighieri wrote The Divine Comedy with its graphic description of...
Boy/Male
Italian Spanish
Enduring. The poet Dante Alighieri wrote The Divine Comedy with its graphic description of...
Surname or Lastname
German (also Gräff), Dutch, and Jewish (Ashkenazic)
German (also Gräff), Dutch, and Jewish (Ashkenazic) : variant of Graf.English : metonymic occupational name for a clerk or scribe, from Anglo-Norman French grafe ‘quill’, ‘pen’ (a derivative of grafer ‘to write’, Late Latin grafare, from Greek graphein).
COMPARABILITY GRAPH
COMPARABILITY GRAPH
Boy/Male
German, Hungarian, Latin
Laurel Trees; From Laurentium; Crowned with Laurels
Boy/Male
Indian
Infinite God of heaven
Boy/Male
Arabic, Muslim, Sindhi
Highland
Female
Croatian
, soul, spirit.
Boy/Male
Hindu, Indian, Marathi
Full of Life; Splendid
Boy/Male
Indian, Punjabi, Sikh
Victory of the Lively
Boy/Male
Danish, German, Swedish
God is Gracious
Girl/Female
English American Greek French Shakespearean
Shining light.
Girl/Female
Yiddish
Bitter.
Girl/Female
Arabic, Bengali, Indian, Muslim, Pakistani
Quiet Night
COMPARABILITY GRAPH
COMPARABILITY GRAPH
COMPARABILITY GRAPH
COMPARABILITY GRAPH
COMPARABILITY GRAPH
n.
An instrument for recording graphically the variations of temperature, or the indications of a thermometer.
a.
Having the faculty of, or characterized by, clear and impressive description; vivid; as, a graphic writer.
adv.
In a graphic manner; vividly.
a.
Pertaining to, containing, derived from, or resembling, graphite.
n.
The quality or power of being compatible or congruous; congruity; as, a compatibility of tempers; a compatibility of properties.
n.
A pen-shaped pointing device used to specify the cursor position on a graphics tablet.
a.
Alt. of Graphitoidal
n.
The quality or state of being graphic.
n.
Compatibility; consistency; fitness; agreement.
n.
See Graphoscope.
n.
An instrument which, when applied over an artery, indicates graphically the movements or character of the pulse. See Sphygmogram.
n.
A crucible; as, a graphite pot; a melting pot.
n.
Anything which represents graphically a succession of events, states, or acts; as, an historical map.
n.
Alt. of Graphicalness
n.
A mineral, a telluride of gold and silver, of a steel-gray, silver-white, or brass-yellow color. It often occurs in implanted crystals resembling written characters, and hence is called graphic tellurium.
a.
Resembling graphite or plumbago.
n.
A chart or graphic representation of the average distribution of rain over the surface of the earth.
a.
Expressing the type, structure, relations, and reactions of a compound; graphic; -- said of formulae. See under Formula.
a.
Alt. of Graphical
n.
An instrument for measuring, and recording graphically, the pressure of the blood in any of the blood vessels of a living animal; -- called also kymographion.