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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
Adjacent subset of an undirected 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 adjacent
Clique_(graph_theory)
Partition of a graph's nodes into cliques
In graph theory, a clique cover or partition into cliques of a given undirected graph is a collection of cliques that cover the whole graph. Generally
Clique_cover
Problem of finding the longest simple path for a given graph
on the clique-width of the graph, so this algorithms is not fixed-parameter tractable. The longest path problem, parameterized by clique-width, is hard
Longest_path_problem
On linear-time algorithms for graph logic
treewidth to clique-width: for every fixed MSO1 property Π {\displaystyle \Pi } , and every fixed bound b {\displaystyle b} on the clique-width of a graph
Courcelle's_theorem
Graph width parameter used in graph theory
Rank-width is closely related to clique-width: k ≤ c ≤ 2 k + 1 − 1 {\displaystyle k\leq c\leq 2^{k+1}-1} , where c {\displaystyle c} is the clique-width and
Rank-width
maximal cliques in G. See also biclique, a complete bipartite subgraph. clique tree A synonym for a block graph. clique-width The clique-width of a graph
Glossary_of_graph_theory
Representation of a graph as a path graph "thickened" by some amount
interval graph with clique number p + 1, then G has a path decomposition of width p whose nodes are given by the maximal cliques of the interval graph
Pathwidth
List of unsolved computational problems
two binary trees be computed in polynomial time? Can graphs of bounded clique-width be recognized in polynomial time? Can one find a simple closed quasigeodesic
List of unsolved problems in computer science
List_of_unsolved_problems_in_computer_science
Problem in graph theory
fixed-parameter tractable for sm-width, another graph width parameter intermediate between treewidth and clique-width. However, under standard assumptions
Maximum_cut
Graph formed by complementation and disjoint union
distance-hereditary graph with diameter at most 2. A cograph is a graph with clique-width at most 2. A cograph is a comparability graph of a series-parallel partial
Cograph
Function in algebraic graph theory
chordal graphs and graphs of bounded clique-width. The latter class includes cographs and graphs of bounded tree-width, such as outerplanar graphs. The
Chromatic_polynomial
rank-width, also has bounded twin-width. The twin-width is at most exponential in the clique-width, and at most doubly exponential in the rank-width. These
Twin-width
Graph whose induced subgraphs preserve distance
graph and therefore of any distance-hereditary graph. Additionally, the clique-width of any distance-hereditary graph is at most three. As a consequence,
Distance-hereditary_graph
Logical formulation of graph properties
fixed MSO1 property can be tested in linear time on graphs of bounded clique-width. The version of this result for graphs of bounded treewidth can also
Logic_of_graphs
Topics referred to by the same term
vertices of any pair of subgraphs formed by removing an edge from the tree. Clique-width of a graph, the minimum number of distinct labels needed to construct
Width_(disambiguation)
Complexity class of problems
Recognizing leaf powers and k-leaf powers Recognizing graphs of bounded clique-width Testing the existence of a simultaneous embedding with fixed edges Testing
NP-intermediate
Graph polynomial generating numbers of matchings
Rotics 2001). The matching polynomial of a graph with n vertices and clique-width k may be computed in time nO(k) (Makowsky et al. 2006). Courcelle, B
Matching_polynomial
Graph of short distances in another graph
its d-th power is the complete graph. If a graph family has bounded clique-width, then so do its d-th powers for any fixed d. Graph coloring on the square
Graph_power
British mathematician
tractable algorithms to approximate the clique-width of graphs (within an exponential bound) and the branch-width of matroids (within a linear bound); and
Paul_Seymour_(mathematician)
Number of matchings in a graph
polynomial (with an exponent that depends linearly on the width) for graphs of bounded clique-width. The Hosoya index can be efficiently approximated to any
Hosoya_index
Intersection graph of unit intervals on the real line
and between treewidth and chordal graphs. A weaker notion of width, the clique-width, may be arbitrarily large on indifference graphs. However, any
Indifference_graph
Austrian computer scientist
on width measures for graphs such as treewidth and clique-width. He showed with coauthors that it is NP-hard to determine whether the clique-width of
Stefan_Szeider
South Korean mathematician (born 1976)
particular for structures and algorithms relating to rank-width, clique-width, and branch-width. He published more than 80 journal papers. He won the Young
Oum_Sang-il
and A. Itai) In graph algorithms, his unifying approach to tree-width and clique-width via model theory, leading to a general theory of graph polynomials
Johann_Makowsky
Bipartite 4-regular graph with 20 nodes and 40 edges
It is also 4-vertex-connected and 4-edge-connected. Its treewidth and clique-width are both 5. The Folkman graph has genus 3: it can be embedded on a triple
Folkman_graph
On chains and antichains in partial orders
number of chains needed to cover all elements. This number is called the width of the partial order. The theorem is named for the mathematician Robert
Dilworth's_theorem
Graph representing leaves of a given tree graph
Gurski, Frank; Wanke, Egon (2009), "The NLC-width and clique-width for powers of graphs of bounded tree-width", Discrete Applied Mathematics, 157 (4): 583–595
Leaf_power
Theorem relating graph minors and topological embeddings
is, we take a k-clique-sum of G1 and G2, then take a k-clique-sum of G3 with the resulting graph, and so on. A graph has tree width at most k if it can
Graph_structure_theorem
Number denoting a graph's closeness to a tree
a tree decomposition of the graph, in terms of the size of the largest clique in a chordal completion of the graph, in terms of the maximum order of a
Treewidth
Canadian mathematician and computer scientist
tree spanners. An explanation of the relationship between tree width and clique-width. Determining the diameter of restricted graph families. Outlining
Derek_Corneil
Indeed, the scope of every original constraint is a clique of the primal graph. This clique remains a clique even after enforcing chordality, as this process
Decomposition method (constraint satisfaction)
Decomposition_method_(constraint_satisfaction)
adjacent in D(H). The matchings in H correspond to the cliques in D(H). Meshulam characterized the widths of a hypergraph H in terms of the properties of D(H)
Width_of_a_hypergraph
Graph where every connected induced subgraph has a universal vertex
(2006), "Characterizations for co-graphs defined by restricted NLC-width or clique-width operations", Discrete Mathematics, 306 (2): 271–277, doi:10.1016/j
Trivially_perfect_graph
Unrelated vertices in graphs
endpoint in S {\displaystyle S} . A set is independent if and only if it is a clique in the graph's complement. The size of an independent set is the number
Independent set (graph theory)
Independent_set_(graph_theory)
Mapping of a graph into a tree
problems on the graph. Tree decompositions are also called junction trees, clique trees, or join trees. They play an important role in problems like probabilistic
Tree_decomposition
Mathematician and computer scientist (born 1948)
1047. MR 1355650. Golumbic, Martin Charles; Rotics, Udi (2000). "On the clique-width of some perfect graph classes". International Journal of Foundations
Martin_Charles_Golumbic
bounded tree width are interpretable using trees, and without predicates over edges this also applies to graphs of bounded clique width. Tree extensions
S2S_(mathematics)
Size of largest complete graph made by contracting edges of a given graph
and edge deletions. The Hadwiger number is also known as the contraction clique number of G or the homomorphism degree of G. It is named after Hugo Hadwiger
Hadwiger_number
{\displaystyle f(t)\in O(t\log(t))} (big O notation) The graphs of bounded clique-width The intersection graphs of scaled and translated copies of any compact
Chi-bounded
Complements of perfect graphs are perfect
vertices. Thus, a clique in the original graph becomes an independent set in the complement and a coloring of the original graph becomes a clique cover of the
Perfect_graph_theorem
Graph of chess rook moves
a clique—a subset of vertices forming a complete graph. The whole rook's graph for an n × m chessboard can be formed from these two kinds of cliques, as
Rook's_graph
Binary operation in graph theory
minor, and k-planar graphs. The clique number of the strong product of any two graphs equals the product of the clique numbers of the two graphs. If two
Strong_product_of_graphs
Methodic assignment of colors to elements of a graph
contains a clique of size k, then at least k colors are needed to color that clique; in other words, the chromatic number is at least the clique number:
Graph_coloring
n + 1 {\displaystyle S^{n+1}} is n {\displaystyle n} . The big-line-big-clique conjecture on the existence of either many collinear points or many mutually
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Measurement of graph sparsity
the arboricity of a graph. Degeneracy is also known as the k-core number, width, and linkage, and is essentially the same as the coloring number or Szekeres–Wilf
Degeneracy_(graph_theory)
Intersection graph of unit disks in the plane
efficiently by using the geometric structure of these graphs, and the maximum clique problem can be solved exactly for these graphs in polynomial time, given
Unit_disk_graph
Intersection graph for intervals on the real number line
be recognized in linear time, and an optimal graph coloring or maximum clique in these graphs can be found in linear time. The interval graphs include
Interval_graph
Generalizations in graph theory
G[V_{Y_{0}}]} contains a clique for every neighbor of Y0 (all edges adjacent to Y0 , that meet at the same vertex of X, form a clique in the line-graph). So
Hall-type theorems for hypergraphs
Hall-type_theorems_for_hypergraphs
Portuguese explorer (c. 1460s – 1524)
the Portuguese government overseas. Turning away from the old Albuquerque clique (now represented by Diogo Lopes de Sequeira), John III looked for a fresh
Vasco_da_Gama
Method of representing systems
Example of a weighted network (weights can also be visualized by giving edges different widths)
Biological_network
Graph that can be embedded in the plane
nodes has at most 8(n-2) maximal cliques, which implies that the class of planar graphs is a class with few cliques. According to Tutte's theorem on Hamiltonian
Planar_graph
Province in North China
museum. In the 2000s, an informal and secretive network known as the Xishan Clique (西山会) emerged linked to Shanxi through birth, family origin, or ancestral
Shanxi
Matroid obtained by restrictions and contractions
matroid, then the width of an e-separation is defined as r(A) + r(B) − r(M) + 1. The width of a decomposition is the maximum width of any of its e-separations
Matroid_minor
49 BC battle, part of Caesar's civil war
Rubicon in January 49 BC, he plunged the Roman Republic into civil war with a clique of Roman senators who were determined to destroy him, under the military
Battle_of_Utica_(49_BC)
History of Brazil before the arrival of Europeans (antiquity-1500)
Brazilian Portuguese). "Emílio Goeldi — Este site está sendo migrado... Clique aqui para acessar novos conteúdos em gov.br/museugoeldi Museu Paraense Emílio
Pre-Cabraline history of Brazil
Pre-Cabraline_history_of_Brazil
Casino resort in Winchester, Nevada, US
Stapleton, Susan (March 5, 2021). "An Ode to All Things Champagne From Clique Hospitality Opens This Summer at Resorts World". Eater. Archived from the
Resorts_World_Las_Vegas
Major river in China
when it migrated, was a topic of controversy in the imperial court. Rival cliques made arguments based on budgetary, technical and strategic criteria. Geographer
Yellow_River
Graphic visual representation of information
Network visualization explores relationships, such as friendships and cliques. Three common types are force-directed layout, arc diagrams, and matrix
Infographic
French tank developed during WWI
downwards to the front. The track consists of thirty-three flat links with a width of thirty-six centimetres. The ground pressure is about 0.75 kg/cm2. As
Schneider_CA1
U.S. state
Virginia's, adopting a township model of government rather than the "courthouse cliques" of Virginia which Carlile criticized, and a compromise demanded by the
West_Virginia
French light tank
training service until 1942) Republic of China (~20 FTs, used by the Fengtian clique and then by the Northeastern Army) Independent State of Croatia (12 former
Renault_FT
Subgraph with contracted edges
in short it establishes that such a graph must have the structure of a clique-sum of smaller graphs that are modified in small ways from graphs embedded
Graph_minor
Mountain gun
Bulgarian GebirgsKanone 06. A GebirgsKanone 06 in use with the Fengtian clique. A GebirgsKanone 06 in Nationalist Chinese service during the Second Sino-Japanese
7.5_cm_GebirgsKanone_06
Canoeing and kayaking involving the descent of waterfalls and slides
through interpersonal communication. This facet of creek boating adds to the clique creation. There are guide books that describe well known runs in detail
Creeking
Subset of a graph's nodes such that all other nodes link to at least one
edge {i, u} for each i ∈ I and u ∈ Si. That is, G is a split graph: I is a clique and U is an independent set. Now if C = {Si : i ∈ D} is a feasible solution
Dominating_set
Capital of Liaoning, China
established on August 8, 1927, with private funding by retired Fengtian clique major general Feng Yong (馮庸, 1901–1981, later re-enlisted as a ROCAF lieutenant
Shenyang
Smallest complete lattice containing a partial order
comparability graph of Q, or a maximal clique in the complement of the comparability graph, so algorithms for the clique problem or the independent set problem
Dedekind–MacNeille_completion
Characterizes the height of any finite partially ordered set
number equals the size of the largest clique. In the comparability graph of a partially ordered set, a clique represents a chain and a coloring represents
Mirsky's_theorem
Puppy Days – animated The Clifton House Mystery - supernatural children's Clique – drama/thriller Clocking Off – drama Clone – comedy Close to Home – sitcom
List of British television programmes
List_of_British_television_programmes
Social clubs in 18th-20th century London
and fortunes entitled them to the entrée anywhere, were excluded by the cliqueism of the lady patronesses; for the female government of 'Almack's' was a
Almack's
Kayles game and clique-forming game: two players alternately select vertices and the induced subgraph must be an independent set (resp. clique). The last to
List of PSPACE-complete problems
List_of_PSPACE-complete_problems
Airport serving Wuxu, Guangxi, China
Nanning Airfield (南寧機場), which had been built in 1929 by the New Guangxi Clique (新桂系軍閥) for military purposes. From 1942 to 1945, the Yongning Airfield
Nanning Wuxu International Airport
Nanning_Wuxu_International_Airport
Zhili clique in the Zhili–Anhui War. Two years later, the Zhili Clique fought off a challenge by Zhang Zuolin and his Manchuria-based Fengtian clique in
History_of_Beijing
Numerical invariant of graphs
the size of the largest clique in this graph. Thus, the tree-depth may alternatively be defined as the size of the largest clique in a trivially perfect
Tree-depth
Subexponential bound in computational complexity
diameter of polytopes in polyhedral combinatorics, or relating the sizes of cliques and independent sets in certain classes of graphs. However, in polyhedral
Quasi-polynomial_growth
Historical region in Gansu Province, China
dynasty and the onset of the Warlord Era. These warlords established the Ma clique, which declared its allegiance to the Kuomintang government in 1927. In
Hexi_Corridor
Physical setting for a learning environment
lessens negative social behavior such as bullying and the development of cliques. The Small Learning Community (SLC) model is structured to provide a more
Learning_space
Threshold of percolation theory models
Interdependent Semantic Spatial Dependency Flow on-Chip Graphs Features Clique Component Cut Cycle Data structure Edge Loop Neighborhood Path Vertex Adjacency
Percolation_threshold
collapsed with the capture of Kashgar by the Hui Muslim forces of the Ma clique on 6 February 1934. On 7 November 1944, uprisings against Chinese rule erupted
Flag_of_East_Turkestan
Libraries donated by Andrew Carnegie
as he wrote: an endowed institution is liable to become the prey of a clique. The public ceases to take interest in it, or, rather, never acquires interest
Carnegie_library
Type of graph related to pursuit–evasion
ordering of the vertices in which the later neighbors of each vertex form a clique) is a valid dismantling order. However, there exist infinite chordal graphs
Cop-win_graph
Method of graph decomposition
separators on minor-closed families of graphs, and characterizing the ends and clique minors of infinite graphs. If G is an undirected graph, and X is a set of
Haven_(graph_theory)
Motor vehicle
suggested as a solution extending the capacity plan in Karis, but certain cliques suspected that SAT tried to benefit from the war. Nessling, who was also
Sisu_S-21
Optimization problem in computer science
precede j in any feasible solution. This problem is NP-hard (reduction by clique) even if all the processing times p j {\displaystyle p_{j}} are equal for
Parallel_task_scheduling
Mathematical system of orderings or sets
{\displaystyle v} that occur later than v {\displaystyle v} in the ordering form a clique. The prefixes of perfect elimination orderings of a chordal graph form an
Antimatroid
Operation in graph theory
prime subgraphs of G {\displaystyle G} along with some star graphs and cliques. Lastly, associate a tree by having one node for each graph in the resulting
Local_complementation
Graph formed by subdivision of triangles
clique is a complete graph on four vertices, formed by choosing any vertex and its three earlier neighbors. Every minimal clique separator (a clique that
Apollonian_network
Type of algorithm
treewidth parameter, and also for cliquewidth. An optimization variant of the k-Clique problem is the Densest k-Subgraph problem (which is a 2-ary Constraint Satisfaction
Parameterized approximation algorithm
Parameterized_approximation_algorithm
19/20th-century British Army officer
him. Murray later wrote (in 1930) "Why did I stay with (this) War Office clique when I knew I was not wanted? I wanted to see Sir John through. I had been
Archibald_Murray
Theatre in Hulme, Manchester, England
that increasingly dominated the dying circuit in the 1950s. The name La Clique is an interesting choice of title, suggesting a show aimed at a select"
Hulme_Hippodrome
German weekly magazine
bourgeoisie. Their function, viewed politically, is not to produce parties but cliques, viewed literarily, not schools but fashions, viewed economically, not
Die_Weltbühne
Street in Rio de Janeiro, Brazil
(August 28, 2015). "O Cine Íris resiste". revista piauí - _pra quem tem um clique a mais. Archived from the original on November 10, 2024. Retrieved April
Rua_da_Carioca
CLIQUE WIDTH
CLIQUE WIDTH
Boy/Male
English
Cliff. Lives at the cliffs. From an English surname and place name. Famous bearer 18th century...
Girl/Female
Tamil
Unique
Girl/Female
Latin American
Only one.
Girl/Female
Tamil
Unique
Male
Spanish
 Pet form of Spanish Enrique, QUIQUE means "home-ruler." Compare with another form of Quique.
Boy/Male
French
Surname or Lastname
English
English : variant spelling of Cliff.
Boy/Male
Shakespearean
The Tragedy of Julius Caesar' A servant to Brutus.
Female
French
Feminine form of French Frédéric, FRÉDÉRIQUE means "peaceful ruler."
Boy/Male
Hindu, Indian
Messenger of God
Girl/Female
Tamil
Unique
Surname or Lastname
English
English : habitational name from any of various places, for example in Shropshire and Cheshire, named Clive, from the dative case of Old English clif ‘slope’, ‘bank’, ‘cliff’ (see Cliff), originally used after a preposition. In some cases the name may be topographical, with the same origin and meaning.
Girl/Female
Tamil
Annjaya | அநà¯à®¨à¯à®œà®¯
Unique
Annjaya | அநà¯à®¨à¯à®œà®¯
Male
English
Cliff Dweller
Girl/Female
Hindu, Indian, Unique
Goddess Lakshmi; Requester; Unique
Boy/Male
Indian
Sweet; Flower; Kind to All
Boy/Male
American, Australian, British, Chinese, Christian, Danish, English, French, German, Jamaican
Cliff; Slope; Bank; Someone who Lived Near a Cliff or a Riverbank; Form of Clifford; From a Steep Cliff
Boy/Male
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sindhi, Telugu
Unique Oneness; Unique
Male
English
English surname transferred to forename use, from the name of various places, derived from Old English clif, CLIVE means "bank, cliff, slope."
Girl/Female
Tamil
Unique
CLIQUE WIDTH
CLIQUE WIDTH
Girl/Female
Hindu, Indian, Sanskrit
Title; Headline; Important
Girl/Female
Arabic, Muslim
The Mother of Kulsum
Boy/Male
Arabic, Muslim
He who is Recommended
Girl/Female
Israeli
Oak tree.
Female
Gaelic
Irish name derived from the Gaelic word blath "flower" with added diminutive suffix, BLÃITHÃN means "little flower."
Girl/Female
Indian
Goddess Durga, Chief of the Goddess, Devee
Boy/Male
Indian, Marathi
One who Sacrifices
Boy/Male
Arabic
Unique; Single
Girl/Female
Afghan, Arabic, Indian, Kannada, Lebanese, Muslim
Glorious; Splendid; Noble; Glory
Female
Native American
Native American Miwok name LILUYE means "singing chicken hawk that soars."
CLIQUE WIDTH
CLIQUE WIDTH
CLIQUE WIDTH
CLIQUE WIDTH
CLIQUE WIDTH
n.
See Cazique.
n.
Same as Silique.
n.
A collection of persons employed to applaud at a theatrical exhibition.
a.
Having oblique angles; as, an oblique-angled triangle.
n.
A genus of naked pteropods. One species (Clione papilonacea), abundant in the Arctic Ocean, constitutes a part of the food of the Greenland whale. It is sometimes incorrectly called Clio.
v. i.
A narrow circle of persons associated by common interests or for the accomplishment of a common purpose; -- generally used in a bad sense.
n.
A light skiff or rowboat used on the Bosporus; also, a Levantine vessel of larger size.
n.
An oblique line.
imp. & p. p.
of Oblique
a.
Of or pertaining to a clique; disposed to from cliques; exclusive in spirit.
n.
See Caique.
p. pr. & vb. n.
of Oblique
n.
An assembly or clique of pedants.
v. i.
To To associate together in a clannish way; to act with others secretly to gain a desired end; to plot; -- used with together.
n.
See Cinque.
v. t.
To excite to action by causing resentment or jealousy; to stimulate; to prick; as, to pique ambition, or curiosity.
n.
The tendency to associate in cliques; the spirit of cliques.
v. i.
To march in a direction oblique to the line of the column or platoon; -- formerly accomplished by oblique steps, now by direct steps, the men half-facing either to the right or left.
v. i.
To deviate from a perpendicular line; to move in an oblique direction.
a.
Oblique; sloping.