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Gluing graphs at complete subgraphs
mathematics, a clique sum (or clique-sum) is a way of combining two graphs by gluing them together at a clique, analogous to the connected sum operation in
Clique-sum
Adjacent subset of an undirected graph
the cliques A, B, and C. The clique-sum is a method for combining two graphs by merging them along a shared clique. Clique-width is a notion of the complexity
Clique_(graph_theory)
Theorem relating graph minors and topological embeddings
clique-sums and vortices. A clique in a graph G is any set of vertices that are pairwise adjacent in G. For a non-negative integer k, a k-clique-sum of
Graph_structure_theorem
Graph family made by joining complete graphs at a universal node
the complete graph Kk at a shared universal vertex. That is, it is a 1-clique-sum of these complete graphs. It has n(k − 1) + 1 vertices and nk(k − 1)/2
Windmill_graph
On forbidden minors in planar graphs
Wagner graph, glued together by clique-sum operations. For instance, K3,3 can be formed in this way as a clique-sum of three planar graphs, each of which
Wagner's_theorem
Size of largest complete graph made by contracting edges of a given graph
number at most four more precisely: they are graphs that can be formed by clique-sum operations that combine planar graphs with the eight-vertex Wagner graph
Hadwiger_number
Problem in graph theory
having the structure of clique-sums of planar graphs and graphs of bounded size. A minor-closed family of graphs has this clique-sum structure exactly when
Maximum_cut
Graph whose peripheral cycles are all triangles
A clique-sum of two graphs is formed by identifying together two equal-sized cliques in each graph, and then possibly deleting some of the clique edges
Strangulated_graph
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
Task of computing complete subgraphs
In computer science, the clique problem is the computational problem of finding cliques (subsets of vertices, all adjacent to each other, also called complete
Clique_problem
Cycle graph with all opposite nodes linked
Klaus Wagner (1937) that graphs with no K5 minor can be formed by using clique-sum operations to combine planar graphs and the Möbius ladder M8; for this
Möbius_ladder
Cubic graph with 8 vertices and 12 edges
Wagner's theorem) that graphs with no K5 minor can be formed by using clique-sum operations to combine planar graphs and the Möbius ladder M8. For this
Wagner_graph
ladder) by clique-sums, operations that glue together subgraphs at cliques of up to three vertices and then possibly remove edges from those cliques. This
Klaus_Wagner
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
Function in algebraic graph theory
if G is a k-clique-sum of G 1 {\displaystyle G_{1}} and G 2 {\displaystyle G_{2}} (i.e., a graph obtained by gluing the two at a clique on k vertices
Chromatic_polynomial
Graph where every edge is in one triangle
smaller locally linear graphs by the following operation, a form of the clique-sum operation on graphs. Let G {\displaystyle G} and H {\displaystyle H} be
Locally_linear_graph
Representation of a graph as a path graph "thickened" by some amount
with a bounded number of apexes and vortices for each component of the clique-sum. An apex is a vertex that may be adjacent to any other vertex in its component
Pathwidth
Set of random variables
\operatorname {cl} (G)} is the set of cliques of G {\displaystyle G} . The definition is equivalent if only maximal cliques are used. The functions φ C {\displaystyle
Markov_random_field
Matroid that can be represented over all fields
co-graphic, using an operation for combining matroids that generalizes the clique-sum operation on graphs. The number of bases in a regular matroid may be computed
Regular_matroid
Matroid obtained by restrictions and contractions
graphs in any minor-closed family can be built up from simpler graphs by clique-sum operations. Some analogous results are also known in matroid theory. In
Matroid_minor
Graph where all long cycles have a chord
induced cycles. Strangulated graphs are graphs that can be formed by clique-sums of chordal graphs and maximal planar graphs. Therefore, strangulated
Chordal_graph
Graph that can be embedded in the plane
the chordal graphs, and are exactly the graphs that can be formed by clique-sums (without deleting edges) of complete graphs and maximal planar graphs
Planar_graph
Representation of a graph's triconnected components
and deleting the two virtual edges. That is, the larger graph is the 2-clique-sum of Gx and Gy. Performing this gluing step on each edge of the SPQR tree
SPQR_tree
Fewest cliques covering a graph's edges
the two values 0 or 1, and a constraint that in each clique of a clique cover the variables sum to at most one. They argue that, for the intersection
Intersection number (graph theory)
Intersection_number_(graph_theory)
Unproven generalization of the four-color theorem
graph that has no K 5 {\displaystyle K_{5}} minor can be decomposed via clique-sums into pieces that are either planar or an 8-vertex Möbius ladder, and
Hadwiger conjecture (graph theory)
Hadwiger_conjecture_(graph_theory)
Extremal graph theory bound on clique-free graph edges
graph that does not contain any ( r + 1 ) {\displaystyle (r+1)} -vertex clique K r + 1 {\displaystyle K_{r+1}} may be formed by partitioning the set of
Turán's_theorem
Period in the history of the Republic of China (1916–1928)
The most powerful cliques were the Zhili clique led by Feng Guozhang, who controlled several northern provinces; the Anhui clique led by Duan Qirui,
Warlord_Era
Upper bound on a graph's Shannon capacity
complement of any graph is sandwiched between the chromatic number and clique number of the graph, and can be used to compute these numbers on graphs
Lovász_number
Describing a family of graphs by excluding certain (sub)graphs
single vertex A finite list of at least 68 billion distinct (1,2,3)-clique sums Graph minor Graphs of spectral radius at most λ {\displaystyle \lambda
Forbidden graph characterization
Forbidden_graph_characterization
Direct sum of uniform matroids
a partition matroid or partitional matroid is a matroid that is a direct sum of uniform matroids. It is defined over a base set in which the elements
Partition_matroid
Degree of connectedness within a graph
Cross-clique centrality of a single node in a complex graph determines the connectivity of a node to different cliques. A node with high cross-clique connectivity
Centrality
Graph cycle which does not separate remaining elements
peripheral cycle is a triangle. They characterize these graphs as being the clique-sums of chordal graphs and maximal planar graphs. Peripheral cycles have also
Peripheral_cycle
Scottish actress (born 1996)
drama Mary Queen of Scots and her television debut in the second series of Clique. She also appeared in theatrical productions of The Selfish Giant and Sylvia
Izuka_Hoyle
On bipartite matching and vertex cover
number and the size of the largest clique are both two while in an independent set the chromatic number and clique number are both one. A graph is perfect
Kőnig's theorem (graph theory)
Kőnig's_theorem_(graph_theory)
graphs of bounded circuit rank, the graphs of bounded pathwidth, the 2-clique-sums of graphs of bounded size, and the k {\displaystyle k} -outerplanar graphs
GNRS_conjecture
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
defined by an excluded induced subgraph, every graph has either a large clique or a large independent set. The Erdős–Mollin–Walsh conjecture on consecutive
List of conjectures by Paul Erdős
List_of_conjectures_by_Paul_Erdős
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)
Graph which partitions into a clique and independent set
{\displaystyle \sum _{i=1}^{m}d_{i}=m(m-1)+\sum _{i=m+1}^{n}d_{i}.} If this is the case, then the m vertices with the largest degrees form a maximum clique in G
Split_graph
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
Complexity class
problems Integer programming Travelling salesman optimization problem Maximum clique Longest simple path Graph coloring; an application: register allocation
NP-hardness
Graph formed by complementation and disjoint union
children, and has size equal to the sum of the children's clique sizes. Thus, by alternately maximizing and summing values stored at each node of the cotree
Cograph
Balanced complete multipartite graph
contain a clique of size r + 1. According to Turán's theorem, the Turán graph has the maximum possible number of edges among all (r + 1)-clique-free graphs
Turán_graph
Geometry problem on tiling by hypercubes
proofs of these results use a reformulation of the problem in terms of the clique number of certain graphs now known as Keller graphs. The related Minkowski
Keller's_conjecture
Machine learning algorithm
The junction tree algorithm (also known as 'Clique Tree') is a method used in machine learning to extract marginalization in general graphs. In essence
Junction_tree_algorithm
Set of computational problems stated by Richard Karp (1973)
Problem) Clique cover Exact cover Hitting set Steiner tree 3-dimensional matching Knapsack (Karp's definition of Knapsack is closer to Subset sum) Job sequencing
Karp's 21 NP-complete problems
Karp's_21_NP-complete_problems
Graph with tight clique-coloring relation
is a graph in which the chromatic number equals the size of the maximum clique, both in the graph itself and in every induced subgraph. In all graphs,
Perfect_graph
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
Concept in graph theory
known as a "Clique graph". The clique graphs have vertices which represent the cliques in the original graph while the edges of the clique graph record
Community_structure
Graph representing edges of another graph
do not have any cliques in common. Therefore, any partition of the graph's edges into cliques would have to have at least one clique for each of these
Line_graph
(or subgraph). A k-clique is a clique of order k. The clique number ω(G) of a graph G is the order of its largest clique. The clique graph of a graph G
Glossary_of_graph_theory
Operation that combines two graphs
{\displaystyle K_{n}=K_{m}+K_{n-m}} (join of two complete graphs whose orders sum to n {\displaystyle n} ). Cograph Cographs are formed by repeated join and
Join_(graph_theory)
Independent set which is not a subset of any other independent set
be maximal-clique irreducible if every maximal clique has an edge that belongs to no other maximal clique, and hereditary maximal-clique irreducible
Maximal_independent_set
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
Equivalence of average-case and expected complexity
many queries are needed for the properties of containing a given tree or clique as a subgraph, of containing a perfect matching, and of containing a Hamiltonian
Yao's_principle
Measure of network community structure
1 {\displaystyle E[J_{vw}]=E\left[\sum _{i=1}^{k_{v}}I_{i}^{(v,w)}\right]=\sum _{i=1}^{k_{v}}E[I_{i}^{(v,w)}]=\sum _{i=1}^{k_{v}}{\frac {k_{w}}{2m-1}}={\frac
Modularity_(networks)
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
Matching_polynomial
Graph coloring where graph elements are assigned sets of colors
linear program computes the "fractional clique number", a relaxation to the rationals of the integer concept of clique number. That is, a weighting of the
Fractional_coloring
Statement in mathematical combinatorics
of its graph-theoretic forms, states that one will find monochromatic cliques in any edge labelling (with colours) of a sufficiently large complete graph
Ramsey's_theorem
directed edges. Variants include the rural postman problem. Clique cover problem Clique problem Complete coloring, a.k.a. achromatic number Cycle rank
List_of_NP-complete_problems
Indian judge
He was also known as a leader in the second generation of the Mylapore clique. He was involved in the prosecution of a partner of the British banking
V._Krishnaswamy_Iyer
large families of graphs; the first family of graphs is obtained from a clique by identifying each of its vertices to a vertex of an arbitrary c-cyclic
Brouwer's_conjecture
2018 concert tour by Beyoncé and Jay-Z
in Love" (contains elements of "Swag Surfin") "Diva" / "Irreplaceable" "Clique" / "Everybody Mad" "Dirt off Your Shoulder" "On to the Next One" "FuckWithMeYouKnowIGotIt"
On_the_Run_II_Tour
theorem Cantor–Bernstein–Schroeder theorem Cayley's formula Cayley's theorem Clique problem (to do) Compactness theorem (very compact proof) Erdős–Ko–Rado theorem
List_of_mathematical_proofs
Node labeling problem in graph theory
as one less than the maximum clique size in a proper interval supergraph of the given graph, chosen to minimize its clique size. For several families of
Graph_bandwidth
American rapper
Tales from the Darkside (with Psychoetry) 2022: Morbid Clique vs. King Gordy (with Morbid Clique) 2023: Kody and the King (with Kody Lee) 2023: Die Slow
King_Gordy
Italian-American organized crime group
also known as the Civella crime family or the Kansas City Mafia or the Clique, is an Italian American Mafia crime family based in Kansas City, Missouri
Kansas_City_crime_family
Tree graph with all nodes within distance 1 from central path
exactly n − k maximal cliques, each containing k + 1 vertices; in a k-tree that is not itself a (k + 1)-clique, each maximal clique either separates the
Caterpillar_tree
Number of edges touching a vertex in a graph
vertex has outdegree exactly 1. By Brooks' theorem, any graph G other than a clique or an odd cycle has chromatic number at most Δ(G), and by Vizing's theorem
Degree_(graph_theory)
Model of computational complexity
computing the sum of its input bits modulo some odd prime p. The k-clique problem is to decide whether a given graph on n vertices has a clique of size k
Circuit_complexity
Clustering and community detection algorithm
the sum of the weights of the edges attached to nodes i {\displaystyle i} and j {\displaystyle j} , respectively; m {\displaystyle m} is the sum of all
Leiden_algorithm
Mathematical game
the same connected component). Similarly, in the clique-forming game, two players must find a clique in the graph. In the normal variant (the last to
Kayles
Distance of a graph from a split graph
partitioned into an independent set (with no edges within this subset) and a clique (having all possible edges within this subset). The splittance is the smallest
Splittance
Theorem in functional analysis
complex) matrix with | ∑ i , j M i j s i t j | ≤ 1 {\displaystyle {\Big |}\sum _{i,j}M_{ij}s_{i}t_{j}{\Big |}\leq 1} for all (real or complex) numbers si
Grothendieck_inequality
Form of artificial neural network
study of retrieval algorithms of sparse messages in networks of neural cliques". COGNITIVE 2014 : The 6th International Conference on Advanced Cognitive
Hopfield_network
Symbols for constants, special functions
Weisstein, Eric W. "Clique". mathworld.wolfram.com. Retrieved 2025-02-07. A clique of a graph G is a complete subgraph of G, and the clique of largest possible
Greek letters used in mathematics, science, and engineering
Greek_letters_used_in_mathematics,_science,_and_engineering
Graph divided into two independent sets
graphs is easy to see (their chromatic number is two and their maximum clique size is also two) but perfection of the complements of bipartite graphs
Bipartite_graph
Graph made from disjoint union of complete graphs
costs (positive and negative) the clique partitioning problem asks for a subgraph that is a cluster graph such that the sum of the costs of the edges of the
Cluster_graph
Spiritual leader of Tibet since 1940
trading family in the small hamlet of Taktser under the control of the Ma clique loyal to the Republic of China as part of Qinghai and on the edge of the
14th_Dalai_Lama
French fast food dish
2018-11-18. "ENQUÊTE : Les Tacos, nouveaux rois du fast-food français". Clique.tv (in French). 2017-01-16. Archived from the original on 2017-11-08. Retrieved
French_tacos
Concept in graph theory and network analysis
\sum _{v\in V}y_{v}\\{\text{s. t.}}&\quad \sum _{v\in V}x_{v}=k\\&\forall v\in V:{\begin{cases}x_{v}+y_{v}\leq 1\\y_{v}\leq \sum _{u\in N(v)}x_{u}\\x_{v}
Group_centrality
the guilt of killing one of Shum Shing-Nam's disciple. Joel Chan Yip Chor-Sum Shum Shing-Nam's second disciple. Ex-love interest of Yip Mung-sik's. Felix
Face_to_Fate
1955 prosecution in the China
government described the group as the Kung Pinmei counterrevolutionary clique and prosecuted them for activities deemed counterrevolutionary. During the
Kung Pinmei counterrevolutionary clique
Kung_Pinmei_counterrevolutionary_clique
Graph of numbers differing by a square
1016/S0378-3758(96)00006-7. Broere, I.; Döman, D.; Ridley, J. N. (1988). "The clique numbers and chromatic numbers of certain Paley graphs". Quaestiones Mathematicae
Paley_graph
x binary } {\displaystyle {\text{maximize }}\left\{\sum _{i=1}^{n}p_{i}x_{i}+\sum _{i=1}^{n}\sum _{j=1,i\neq j}^{n}P_{ij}x_{i}x_{j}:x\in X,x{\text{ binary}}\right\}}
Quadratic_knapsack_problem
Mathematically incorrect slogan
line of thought is a nightmare world in which the Leader, or some ruling clique, controls not only the future, but the past. If the Leader says of such
2_+_2_=_5
Method of partitioning data points into groups based on their similarity
}{\operatorname {maximize} }}&&\sum _{e\in E\setminus \delta (\Pi )}c_{e}\;.\end{aligned}}} This formulation is also known as the clique partitioning problem. It
Correlation_clustering
Grouping a set of objects by similarity
results and just provide the grouping information. Graph-based models: a clique, that is, a subset of nodes in a graph such that every two nodes in the
Cluster_analysis
American socialite and Titanic survivor (1893–1940)
New York social life, she was immediately adopted by the Junior League, a clique of debutantes. She appeared in several New York society plays and attracted
Madeleine_Astor
Generalization of metric spaces
) {\displaystyle \delta (A)} is defined for any finite A as the largest clique of A, then ( X , δ ) {\displaystyle (X,\delta )} is a diversity. Bryant
Diversity_(mathematics)
Solving an optimization problem with a quadratic objective function
associated with G has a maximum that is function of the clique number of G. Computing the clique number of a graph is a well-known NP-hard problem; hence
Quadratic_programming
History of intellectualism in China
intellectuals allied themselves with cliques within the government to lend support to the policies of that clique. With the abolition of the civil service
Chinese_intellectualism
Problem of determining if a Boolean formula could be made true
problem. An example of a problem where this method has been used is the clique problem: given a CNF formula consisting of c clauses, the corresponding
Boolean satisfiability problem
Boolean_satisfiability_problem
where costs vary Cliques Bron–Kerbosch algorithm: a technique for finding maximal cliques in an undirected graph MaxCliqueDyn maximum clique algorithm: find
List_of_algorithms
Process by which people befriend similar people
locally, for example at the node/neighborhood level and at the group-size or clique level (within-group vs across-group homophily), which can lead to different
Homophily
American supernatural thriller television series
They soon learn that the officer is a "Protector", a member of an NYPD clique obsessed with the popular cop show Justice Served. The show's producer,
Evil_(American_TV_series)
Subfield of computational complexity theory
for proving lower bounds on dynamic problems. The k-Clique hypothesis concerns detecting k-cliques in graphs; the best known algorithm uses fast matrix
Fine_grained_complexity
Measure of centrality in a network based on nodal influence
S2CID 121768822. Charles H. Hubbell (1965). "An input-output approach to clique identification". Sociometry. 28 (4): 377–399. doi:10.2307/2785990. JSTOR 2785990
Katz_centrality
State of being organized by, or advocating for, tribes or tribal lifestyles
for tribalism. Amity-enmity complex Anarcho-primitivism Chauvinism Clan Clique Cult Communitarianism Community Engaged theory Esprit de corps Ethnocentrism
Tribalism
Theorem in geometry about convex sets
a i x i = 0 , ∑ i = 1 d + 2 a i = 0 , {\displaystyle \sum _{i=1}^{d+2}a_{i}x_{i}=0,\quad \sum _{i=1}^{d+2}a_{i}=0,} because there are d + 2 unknowns
Radon's_theorem
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