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The term subgraph can refer to: The security-focused Linux-based Subgraph operating system, see Subgraph (operating system) Subgraph of a function, see...

theory, a Sachs subgraph of a given graph is a subgraph in which all connected components are either single edges or cycles. These subgraphs are named after...

In metric graph theory, a convex subgraph of an undirected graph G is a subgraph that includes every shortest path in G between two of its vertices. Thus...

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...

Forbidden subgraph may refer to: Forbidden subgraph characterization, in graph theory, a characterization of a family of graphs by exclusion of graphs...

states that a finite graph is planar if and only if it does not contain a subgraph that is a subdivision of K 5 {\displaystyle K_{5}} (the complete graph...

V W X Y Z See also References Square brackets [ ] G[S] is the induced subgraph of a graph G for vertex subset S. Prime symbol ' The prime symbol is often...

In graph theory and computer science, a dense subgraph is a subgraph with many edges per vertex. This is formalized as follows: let G = (V, E) be an undirected...

In theoretical computer science, the subgraph isomorphism problem is a computational task in which two graphs G {\displaystyle G} and H {\displaystyle...

computer science, a maximum common subgraph may mean either: Maximum common induced subgraph, a graph that is an induced subgraph of two given graphs and has...

In extremal graph theory, the forbidden subgraph problem is the following problem: given a graph G {\displaystyle G} , find the maximal number of edges...

First, a large planar subgraph is found within the given graph. Then, the remaining edges that are not already part of this subgraph are added back one at...

from the family which contain any of these forbidden graphs as (induced) subgraph or minor. A prototypical example of this phenomenon is Kuratowski's theorem...

graph theory, induced subgraph isomorphism is an NP-complete decision problem that involves finding a given graph as an induced subgraph of a larger graph...

Subgraph OS was a Debian-based project designed to be resistant to surveillance and interference by sophisticated adversaries over the Internet. It has...

In graph theory, a uniconnected subgraph is a directed graph that has at most one path between any pair of vertices. Given a directed graph G = ( V , E...

which every such colouring contains at least one single-coloured complete subgraph on four coplanar vertices? In 1971, Graham and Rothschild proved the Graham–Rothschild...

graph is the set of its even-degree spanning subgraphs, or the set of their edge sets. This set of subgraphs can be described algebraically as a vector...

"Google of blockchains," The Graph allows developers to create and query Subgraphs (open APIs) and Substreams (real-time data pipelines), and Token API (real-time...

hereditary for subgraphs, which means that a graph has the property if and only if all subgraphs have it too. Finding maximal subgraphs of a certain kind...

computer science, a maximum common induced subgraph of two graphs G and H is a graph that is an induced subgraph of both G and H, and that has as many vertices...

Unsolved problem in mathematics Are graphs uniquely determined by their subgraphs? More unsolved problems in mathematics In graph theory, informally, the...

these edges from the graph breaks all of the cycles, producing an acyclic subgraph of the given graph, often called a directed acyclic graph. A feedback arc...

component of an undirected graph is a connected subgraph that is not part of any larger connected subgraph. The components of any graph partition its vertices...

isomorphic to both a subgraph of G {\displaystyle G} and a subgraph of G ′ {\displaystyle G'} . The maximum common edge subgraph problem on general graphs...

induced subgraphs. Roughly speaking, instead of finding a monochromatic subgraph, we are now required to find a monochromatic induced subgraph. In this...

which every non-empty subgraph has at least one vertex of degree at most k {\displaystyle k} . That is, some vertex in the subgraph touches k {\displaystyle...

hereditary family of graphs, they can be characterized by forbidden induced subgraphs. The unit distance graphs include the cactus graphs, the matchstick graphs...

build the Higman–Sims graph, which has the Hoffman-Singleton graph as a subgraph. Let G {\displaystyle G} be the set Z 2 × Z 5 × Z 5 {\displaystyle \mathbb...

complementary subset is as large as possible. Equivalently, one wants a bipartite subgraph of the graph with as many edges as possible. There is a more general version...

theorem: A finite graph is planar if and only if it does not contain a subgraph that is a subdivision of the complete graph K5 or the complete bipartite...

That is, it is a minimal set of cycles that allows every even-degree subgraph to be expressed as a symmetric difference of basis cycles. A fundamental...

set of vertices, then an X-flap is a nonempty connected component of the subgraph of G formed by deleting X. A haven of order k in G is a function β that...

induced subgraphs with many vertices. An induced subgraph of even degree can be found with at least half of the vertices, and an induced subgraph of odd...

connected subgraph (or equivalently every connected induced subgraph) has at least one vertex whose color is unique: no other vertex in the same subgraph has...

k} disjoint connected subgraphs of G {\displaystyle G} such that each subgraph is connected by an edge to each other subgraph. Contracting the edges...

and H is a graph formed by combining G and H that has applications to subgraph isomorphism. It is one of several different kinds of graph products that...

federal prison, he was involved[how?] in some security projects, such as the Subgraph OS in 2017. On August 8, 2014, he and Ladar Levison presented the Dark...

In graph theory, the neighbourhood of a vertex v in a graph G is the subgraph of G induced by all the vertices that are connected to v by an edge (vertices...

even number of vertices. Find a minimum-weight perfect matching M in the subgraph induced in G by O. Combine the edges of M and T to form a connected multigraph...

that can be included in an undirected graph that does not have a complete subgraph of a given size. It is one of the central results of extremal graph theory...

bipartite graphs are perfect. Line graphs are characterized by nine forbidden subgraphs and can be recognized in linear time. Various extensions of the concept...

vertices in previous subgraphs, and each cycle other than the first in the sequence having exactly one vertex in previous subgraphs. For instance, the graph...

variables into a subgraph that repeat together, and a number is drawn on the plate to represent the number of repetitions of the subgraph in the plate. The...

a claw-free graph is a graph that does not have a claw as an induced subgraph. A claw is another name for the complete bipartite graph K 1 , 3 {\displaystyle...

number of vertices and edges) and local (such as the existence of specific subgraphs), and problems in extremal graph theory can often be formulated as optimization...

adjacent. That is, a clique of a graph G {\displaystyle G} is an induced subgraph of G {\displaystyle G} that is complete. Cliques are one of the basic concepts...

G is a spanning subgraph, i.e., a subgraph that has the same vertex set as G. A k-factor of a graph is a spanning k-regular subgraph, and a k-factorization...

graph is a graph in which every isomorphism between two of its induced subgraphs of at most k vertices can be extended to an automorphism of the whole...

perfect graph is a graph with the property that in each of its induced subgraphs the size of the maximum independent set equals the number of maximal cliques...

other words, the thickness of a graph is the minimum number of planar subgraphs whose union equals to graph G. Thus, a planar graph has thickness one...

graph that has a given number of vertices and has no complete bipartite subgraphs of a given size? More unsolved problems in mathematics The Zarankiewicz...

acyclic graph (a directed graph without directed cycles) is unique and is a subgraph of the given graph. However, uniqueness fails for graphs with (directed)...

strongly connected components of a directed graph form a partition into subgraphs that are strongly connected themselves. It is possible to test the strong...

graph is an induced subgraph of the Rado graph, and can be found as an induced subgraph by a greedy algorithm that builds up the subgraph one vertex at a...

tree exists in subgraph composed solely with edges in smaller edges set, it then computes an MBST in the subgraph, an MBST of the subgraph is exactly an...

{t}{\varepsilon }})}n^{O(1)}} time. It is known that the Strongly Connected Steiner Subgraph problem is W[1]-hard parameterized by the number k of terminals, and also...

an overfull subgraph, an overfull graph that is a subgraph, immediately follows. An alternate, stricter definition of an overfull subgraph S of a graph...

weighted graph is a connected subgraph, without cycles, for which the sum of the weights of all the edges in the subgraph is minimal. For a disconnected...

disjoint union. The topology on this space is called the graph topology. A subgraph of a graph X {\displaystyle X} is a subspace Y ⊆ X {\displaystyle Y\subseteq...

(a polynomial number of times). The subgraph isomorphism problem of determining whether graph G contains a subgraph that is isomorphic to graph H. Turing...

(subsets of vertices, all adjacent to each other, also called complete subgraphs) in a graph. It has several different formulations depending on which...

1109/IEMBS.1999.803869. ISBN 0-7803-5674-8. Paul A. Catlin (1977). "Embedding subgraphs under extremal degree conditions" (PDF). Congressus Numerantium. 19: 136–45...

transversal from a graph leaves a bipartite graph as the remaining induced subgraph. A given n {\displaystyle n} -vertex graph G {\displaystyle G} has an odd...

{\displaystyle G} is connected, this forest must be a single tree; it need not be a subgraph of G {\displaystyle G} , but if it is, it is a Trémaux tree for G {\displaystyle...

distances in any connected induced subgraph are the same as they are in the original graph. Thus, any induced subgraph inherits the distances of the larger...

it applies to the subgraph isomorphism problem (an NP-complete problem), where it yields polynomial time algorithms when the subgraph pattern that it is...

simple cycle, or (for nontrivial cacti) in which every block (maximal subgraph without a cut-vertex) is an edge or a cycle. Cacti are outerplanar graphs...

a graph that is an isometric subgraph of a hypercube. In other words, a partial cube can be identified with a subgraph of a hypercube in such a way that...

subgraphs of G that all touch each other: for every pair of disjoint subgraphs, there must exist an edge in G that has one endpoint in each subgraph....

Network motifs are recurrent and statistically significant subgraphs or patterns of a larger graph. All networks, including biological networks, social...

these properties have efficient algorithms for problems including the subgraph isomorphism problem and model checking for the first order theory of graphs...

subgraphs and cross edges. Using maximum flow algorithms to obtain the minimum cut of the constructed graph. The minimum cut within each GTW subgraph...

induced from the bigger structure. Subgraphs are an example where the distinction matters, and the term "subgraph" does indeed refer to weak substructures...

be removed to separate the remaining nodes into two or more isolated subgraphs. It is closely related to the theory of network flow problems. The connectivity...

theorem, a finite graph is planar if and only if it does not contain a subgraph that is a subdivision of K5, or K3,3, the utility graph. There are five...

infinite graph to the same problem on its finite subgraphs. It states that, when all finite subgraphs can be colored with c {\displaystyle c} colors, the...

found as subgraphs of the digraph of a relation are called concepts. When a lattice is formed by taking meets and joins of these subgraphs, the relation...

array { 5 , 4 , 2 ; 1 , 1 , 4 } {\displaystyle \{5,4,2;1,1,4\}} . It is a subgraph of the Hoffman–Singleton graph. Brouwer, A. E.; Cohen, A. M.; Neumaier...

maximal acyclic subgraph of the orientation, and then coloring each vertex by the length of the longest path in the chosen subgraph that ends at that...

directed edge: Determining whether for graphs G and H, H is homeomorphic to a subgraph of G, is an NP-complete problem. The reverse operation, smoothing out or...

strong hardness of approximation results for finding complete bipartite subgraphs. In 2010, Sanjeev Arora, Boaz Barak and David Steurer found a subexponential...

homomorphism problem Graph partition into subgraphs of specific types (triangles, isomorphic subgraphs, Hamiltonian subgraphs, forests, perfect matchings) are...

one directed path between any two vertices, or equivalently in which the subgraph reachable from any vertex induces an undirected tree, or a partially ordered...

for every graph H, the property of not containing H as an induced subgraph subgraph is testable. In 2005, Alon and Shapira showed that any monotone graph...

removal splits G into two (possibly disconnected) subgraphs with at most 2n⁄3 vertices per subgraph. Even stronger, for any fixed H, H-minor-free graphs...

forbidden induced subgraphs. A perfect graph is an undirected graph with the property that, in every one of its induced subgraphs, the size of the largest...

generated by GPT-5.6, its large language model. A cycle is a connected subgraph all of whose vertices have degree 2. In particular, a single edge, traversed...

which is to be matched to a subgraph in the complete state, and a replacing graph, which will replace the matched subgraph. Formally, a graph rewriting...

Unsolved problem in mathematics Do the graphs with a fixed forbidden induced subgraph necessarily have large cliques or large independent sets? More unsolved...

Unsolved problem in mathematics Are digraphs uniquely determined by their subgraphs and some in-degree data? More unsolved problems in mathematics The reconstruction...

(sometimes known as a 2-connected component) is a maximal biconnected subgraph. Any connected graph decomposes into a tree of biconnected components called...

quasi-random: the number of times each possible constant-order graph occurs as a subgraph of a Paley graph is (in the limit for large q) the same as for random graphs...

The Maximum Degree-and-Diameter-Bounded Subgraph problem (MaxDDBS) is a problem in graph theory. Given a connected host graph G {\displaystyle G} , an...

drawing, a subhamiltonian graph is a subgraph of a planar Hamiltonian graph. A graph G is subhamiltonian if G is a subgraph of another graph aug(G) on the same...

minimum spanning tree is, in fact, a minimum-cost subgraph connecting all vertices, since if a subgraph contains a cycle, removing any edge along that cycle...

field of graph theory, a spanning tree T of an undirected graph G is a subgraph that is a tree which includes all of the vertices of G. In general, a graph...

graph into two subgraphs of maximum degree two. The paths and even cycles of each subgraph may be colored with two colors per subgraph. After this step...

induced subgraph has at least two vertices with the same neighbourhoods. A cograph is a graph in which every nontrivial connected induced subgraph has a...

single row or column (the clique number of the induced subgraph). This class of induced subgraphs are a key component of a decomposition of perfect graphs...

the covering relation of the reachability relation ≤ of the DAG. It is a subgraph of the DAG, formed by discarding the edges u → v for which the DAG also...

where the parentheses demark the subsequences within the two subgraphs of the partition. Fibonacci cubes with an even number of nodes greater...

rise to the Möbius–Kantor configuration. The Möbius–Kantor graph is a subgraph of the four-dimensional hypercube graph, formed by removing eight edges...

even subgraph of a graph (also known as an Eulerian subgraph) is one in which every vertex is incident with an even number of edges. Such subgraphs are...

an induced path in an undirected graph G is a path that is an induced subgraph of G. That is, it is a sequence of vertices in G such that each two adjacent...

goal is to partition the edges of a given graph into two triangle-free subgraphs. It is NP-complete but fixed-parameter tractable on graphs of bounded...

every graph in a certain class contains some subgraph from a specified list. The presence of the desired subgraph is then often used to prove a coloring result...

G_{1}+G_{2}} is connected ( K p 1 , p 2 {\displaystyle K_{p_{1},p_{2}}} is a subgraph). The chromatic number of the join satisfies: χ ( G 1 + G 2 ) = χ ( G 1...

matching is a special kind of subgraph useful in many computational contexts. Given a graph G, a matching is a subgraph where no two edges share a vertex...

limited-depth minor is a restricted form of a graph minor in which the subgraphs that are contracted to form the minor have small diameter. Shallow minors...

graph theory, a critical graph is an undirected graph all of whose proper subgraphs have smaller chromatic number. In such a graph, every vertex or edge is...

to search for an optimal subgraph within a large graph. The controller is trained with policy gradient to select a subgraph that maximizes the validation...

{\displaystyle r} -subgraphs is strictly less than 1 {\displaystyle 1} . The number of monochromatic r {\displaystyle r} -subgraphs in this random coloring...

Gluing graphs at complete subgraphs...

graph G1 isomorphic to graph G2? Subgraph Isomorphism: Is graph G1 isomorphic to a subgraph of graph G2? The Subgraph Isomorphism problem is NP-complete...

k-constructible graph as a subgraph. Equivalently, every k-critical graph (a graph that requires k colors but for which every proper subgraph requires fewer colors)...

randomly select any two subgraphs - so what are the criteria for selecting? We need to choose graphs such that: the two subgraphs have no edges in common...

theory and geometric topology that each describe the cliques (complete subgraphs) of an undirected graph. The clique complex X(G) of an undirected graph...

subgraph. The semantics are: The blank page denotes Truth; Letters, phrases, subgraphs, and entire graphs may be True or False; To enclose a subgraph...

Applications of matroid parity algorithms include finding large planar subgraphs and finding graph embeddings of maximum genus. Matroid parity algorithms...

every ( 2 n − 2 ) {\displaystyle (2n-2)} -vertex tournament contain as a subgraph every n {\displaystyle n} -vertex oriented tree? More unsolved problems...

i-vertex clique but that does contain all Ki-free finite graphs as induced subgraphs. For instance, G3 is a triangle-free graph that contains all finite triangle-free...

S2CID 16399588. Small Molecule Subgraph Detector (SMSD)— a Java-based software library for calculating Maximum Common Subgraph (MCS) between small molecules...

as an induced subgraph the Grötzsch graph, the smallest triangle-free four-chromatic graph, and every four-chromatic induced subgraph of the Clebsch...

in 1958. Let G be a simple, undirected graph. A spanning tree of G is a subgraph of G that is a tree with the same vertex set as G. The Laplacian matrix...

conductance, which measures the likelihood that a random walk on a subgraph will exit that subgraph. Informal networks and organizational crises: An experimental...

that a minimal separator is not the same thing as a minimal separating subgraph.) Dirac used this characterization to prove that chordal graphs are perfect...

conditions: G is connected and has n − 1 edges. G is connected, and every subgraph of G includes at least one vertex with zero or one incident edges. (That...

rate) for the so-called VC subgraph classes. For a function f : X → R {\displaystyle f:{\mathcal {X}}\to \mathbf {R} } the subgraph is a subset of X × R {\displaystyle...

feasible solution. This technique has given PTASs for the following problems: subgraph isomorphism, maximum independent set, minimum vertex cover, minimum dominating...

size of the maximum clique, both in the graph itself and in every induced subgraph. In all graphs, the chromatic number is greater than or equal to the size...

have high arboricity, and graphs with high arboricity must have a dense subgraph. In more detail, as any n-vertex forest has at most n − 1 edges, the arboricity...

that both the resulting edge-monochromatic red and blue vertex-spanning subgraphs are copies of the Ljubljana graph. These two copies contain exactly the...

the graph into subgraphs using separators, small subsets of vertices the removal of which allows the graph to be partitioned into subgraphs with at most...

Finding a graph with the fewest vertices that does not appear as an induced subgraph of a given graph can be solved in time n O ( log ⁡ n ) {\displaystyle n^{O(\log...

language, which converts readable policies into seccompb-bpf bytecode Subgraph OS uses seccomp-bpf Flatpak uses seccomp for process isolation Bubblewrap...

contain an induced K 1 , 6 {\displaystyle K_{1,6}} subgraph. Infinitely many other forbidden induced subgraphs are known. The number of unit disk graphs on...

homomorphism. There is a connection between homomorphism densities and subgraph densities, which is elaborated on below. The edge density of a graph G...

vice versa. Any induced subgraph of the complement graph of a graph G is the complement of the corresponding induced subgraph in G. An independent set...

of a given subgraph, then all of the copies can be eliminated by removing a small number of edges. The special case in which the subgraph is a triangle...

by a path that stays entirely within D. That is, D induces a connected subgraph of G. Every vertex in G either belongs to D or is adjacent to a vertex...

graph contains an edge (v,w). The neighborhood of a vertex v is an induced subgraph of the graph, formed by all vertices adjacent to v. The degree of a vertex...

regular complex polytope, 3{6}2 or or , represents the Cayley diagram for the binary tetrahedral group, with each red and blue triangle a directed subgraph....

when the graph is squared. The half-square of a bipartite graph G is the subgraph of G2 induced by one side of the bipartition of G. Map graphs are the half-squares...

subnormal sub-group Connected component (graph theory), a maximal connected subgraph Connected component (topology), a maximal connected subspace of a topological...

problem is a special case of the subgraph isomorphism problem, which asks whether a given graph G contains a subgraph that is isomorphic to another given...

with n vertices in d-dimensional space, a set of edges that defines a subgraph with k degrees of freedom has matroid rank dn − k. A set of edges is independent...

Fulkerson Prize. A perfect graph is a graph in which, for every induced subgraph, the size of the maximum clique equals the minimum number of colors in...

undirected graph G {\displaystyle G} has a vertex for every clique (complete subgraph) of G {\displaystyle G} ; two vertices of κ ( G ) {\displaystyle \kappa...

problem in mathematics Does forbidding both a tree and a clique as induced subgraphs produce graphs of bounded chromatic number? More unsolved problems in...

has no odd cycle as a subgraph, and a graph is perfect if and only if it has no odd cycle or its complement as an induced subgraph. The bipartite graphs...

Turán graphs are complement graphs of cluster graphs, with all complete subgraphs of equal or nearly-equal size. The locally clustered graph (graphs in...

induced subgraph that connects the query vertices and minimizes the sum of shortest path distances among all pairs of vertices in the subgraph. In combinatorial...

four edges are removed from the graph (red edges, right), the remaining subgraph becomes triangle-free, and more strongly bipartite (as shown by the blue...

v_{5}} are colored with colors 1, 2, 3, 4, 5 respectively. Now consider the subgraph G 1 , 3 {\displaystyle G_{1,3}} of G ′ {\displaystyle G'} consisting of...

arbitrarily large finite subgraphs with any density less than its upper density, and does not have arbitrarily large finite subgraphs with density greater...

induced subgraph of the Clebsch graph, and every triangle-free four-chromatic P 6 {\displaystyle P_{6}} -free graph is likewise an induced subgraph of the...

programs, which is to say, the minimal subgraphs that make the CFG of a program non-structured. These subgraphs have a very good description in natural...

The Highly Connected Subgraphs (HCS) clustering algorithm (also known as the HCS algorithm, and other names such as Highly Connected Clusters/Components/Kernels)...

1-NNG. k-NNGs obey a separator theorem: they can be partitioned into two subgraphs of at most n(d + 1)/(d + 2) vertices each by the removal of O(k1/dn1 − 1/d)...

iff for every U ⊂ V ( G ) {\displaystyle U\subset V(G)} , the induced subgraph G [ U ] {\displaystyle G[U]} has at most t ( | U | − 1 ) {\displaystyle...

k-components) are always a subgraph of a k-core, although a k-core is not always k-cohesive. A k-core is simply a subgraph in which all nodes have at...

distinguished from an unrelated meaning of "bridge" in graph theory, a subgraph separated from the rest of the graph by a specified subset of vertices;...

one at a time to an induced subgraph, and finding the solution to the induced subgraph, as follows: Start with a subgraph induced by a vertex set S of...

{\displaystyle u} . The subgraph isomorphism problem for a fixed subgraph H {\displaystyle H} asks whether H {\displaystyle H} appears as a subgraph of a larger graph...

in terms of the maximum order of a bramble, a collection of connected subgraphs that all touch each other. Treewidth is commonly used as a parameter in...

probabilistic logic, arithmetic. relational models C++, Scheme, Python no subgraph isomorphism OpenSMT Linux, Mac OS, Windows GPLv3 partial v2.0 Yes empty...

connected components of induced subgraphs of the given graph, parameterized by the number of vertices in the subgraph. Knot polynomial Shi, Yongtang;...

if each of these edges is replaced by an appropriate subgraph, such that each replacement subgraph is itself asymmetric and two replacements are isomorphic...

including graph and hypergraph packing problems, random graphs and random subgraphs, and the relations between graph parameters including graph bandwidth...

characterized in terms of their forbidden induced subgraphs: a graph is split if and only if no induced subgraph is a cycle on four or five vertices, or a pair...

considered to be making a subgraph in which it is challenging to find a route between two particular nodes. If the subgraph is not connected, then there...

be v1, v2, . . . , vn. The Mycielski graph μ(G) contains G itself as a subgraph, together with n+1 additional vertices: a vertex ui corresponding to each...

have the diamond graph or a cycle of four or more vertices as an induced subgraph; that is, they are the diamond-free chordal graphs. They are also the Ptolemaic...

bp in the Delaunay triangulation since the nearest neighbor graph is a subgraph of the Delaunay triangulation. The Delaunay triangulation is a geometric...

disconnect it, then it cannot be of class one. By the handshaking lemma, the subgraphs on either side of the bridge have an odd number of vertices each. Whichever...

hyperlinks among those pages form a focused subgraph. The HITS computation is performed only on this focused subgraph. According to Kleinberg the reason for...

homomorphism r from a graph G to a subgraph H of G such that r(v) = v for each vertex v of H. In this case the subgraph H is called a retract of G. A core...

from the node v, and reporting all strongly connected components of that subgraph. When each node finishes recursing, if its lowlink is still set to its...

hyperbolic plane, and finding a graph with few vertices that is not an induced subgraph of a given graph. The exponential time hypothesis also implies that no...

particular on distances in graphs and on the color-coding technique for subgraph isomorphism. With Howard Karloff, he is the namesake of the Karloff–Zwick...

Łuczak) is known when P is monotone with respect to the subgraph ordering (meaning that if A is a subgraph of B and B satisfies P, then A will satisfy P as well)...

asks for a tree of minimum cost that has exactly k vertices and forms a subgraph of a larger graph. It is also called the k-MST or edge-weighted k-cardinality...

be a probability parameter to be chosen later, and construct a random subgraph H of G by allowing each vertex of G to lie in H independently with probability...

Block, in graph theory, is a biconnected component, a maximal biconnected subgraph of a graph Aschbacher block of a finite group Block design, a kind of set...

{\displaystyle c} . The capacity constraint ensures that all subtrees (maximal subgraphs connected to the root by a single edge) incident on the root node r {\displaystyle...

(where the O invokes big O notation) can partition the graph into disjoint subgraphs each of which has at most ⁠ 2 n / 3 {\displaystyle 2n/3} ⁠ vertices. A...

subgraph. The semantics are: The blank page denotes Truth; Letters, phrases, subgraphs, and entire graphs may be True or False; To enclose a subgraph...

computer science. She has published research on the use of maximum common subgraphs and Petri nets to model problems in biology, and on the prediction of...

way that u and v belong to different K6 subgraphs of the product. The Schläfli graph has a total of 36 subgraphs of this form, one of which consists of...

{\log k}})} ⁠ edges. If a graph G has Hadwiger number k, then all of its subgraphs also have Hadwiger number at most k, and it follows that G must have degeneracy...

has its edges partitioned into complete bipartite subgraphs, at least k {\displaystyle k} subgraphs are needed. Equivalently, their conjecture states...

of circles that satisfies the theta-graph property mentioned above.) A subgraph or edge set whose circles are all in B (and which contains no half-edges)...

which is an induced subgraph of the next graph in the sequence, with the property that every graph in the family is an induced subgraph of a graph in the...

list of 9 forbidden induced subgraphs. (See the article on line graphs.) No characterization by forbidden induced subgraphs is known of line graphs of...

algorithms, for finding specific sub-structures in a given graph — such as subgraphs, paths, circuits, and so on. Examples include Dijkstra's algorithm, Kruskal's...

(obsolete) name complexity. The complement of such a set is a balanced subgraph of Σ with the most possible edges. Finding the frustration index is an...

to its number of vertices) must contain a k {\displaystyle k} -fan as a subgraph. More specifically, this is true for an n {\displaystyle n} -vertex graph...

with boundary. Boundary (graph theory), the vertices of edges between a subgraph and the rest of a graph Boundary (chain complex), its abstractization in...

edges of G into two subgraphs, and the width of the decomposition is the maximum number of shared vertices of any pair of subgraphs formed in this way...

Subgraph...

Petersen graphs, by finding large planar subgraphs of these graphs, using Grinberg's theorem to show that these subgraphs are non-Hamiltonian, and concluding...

preorders defined on graphs: A graph property P is hereditary if every induced subgraph of a graph with property P also has property P. For instance, being a perfect...

It is not true that every induced subgraph of a cop-win graph is cop-win. However, certain special induced subgraphs do remain cop-win. Nowakowski & Winkler...

The 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...

her thesis was Approximation Algorithms for Planar and Highly Connected Subgraphs. Her research focus lies in the research of combinatorial optimization...

that contains every finite (or at-most-countable) graph as an induced subgraph. A universal graph of this type was first constructed by Richard Rado and...

graph theory, a partial k-tree is a type of graph, defined either as a subgraph of a k-tree or as a graph with treewidth at most k. Many NP-hard combinatorial...

a union of disjoint (finite) simple cycles if and only if every finite subgraph of G can be extended (by including more edges and vertices from G) to a...

coloring. Perfect graphs are defined as graphs in which, for every induced subgraph, the chromatic number (minimum number of colors in a coloring) equals the...

graphs can be applied to dense graphs like counting the copies of a given subgraph within graphs. Endre Szemerédi proved the lemma over bipartite graphs for...

g Metric tensor (general relativity) Gluon H represents: an arbitrary subgraph an arbitrary subgroup a Hilbert space the unit henry of magnetic inductance...

labeled bipartite subgraph of a complete bipartite graph to a given degree sequence. The hitchcock problem asks for such a subgraph minimizing the sum...

satisfied by X. The Bernoulli bond percolation subgraph of a graph G at parameter p is a random subgraph obtained from G by deleting every edge of G with...

engine, for performing graph and hypergraph pattern matching (isomorphic subgraph discovery). This generalizes the idea of a structured query language (SQL)...

j ) = c ( i , j ) {\displaystyle y(i)+y(j)=c(i,j)} . Let us denote the subgraph of tight edges by G y {\displaystyle G_{y}} . The cost of a perfect matching...

A skein in a graph ⁠ G {\displaystyle G} ⁠ is a subgraph of ⁠ G {\displaystyle G} ⁠ that is the union of a collection of paths between two distinct vertices...

The symmetries of a regular chiliagon. Light blue lines show subgroups of index 2. The 4 boxed subgraphs are positionally related by index 5 subgroups....

kinds of edges make up the entire strong product. Every planar graph is a subgraph of a strong product of a path and a graph of treewidth at most six. This...

coalition can improve upon Core (graph theory), the homomorphically minimal subgraph of a graph Core (group theory), an object in group theory Core of a triangulated...

Subgraph of Delaunay triangulation...

isomorphic graph. Every maximal planar graph, other than K4 = W4, contains as a subgraph either W5 or W6. There is always a Hamiltonian cycle in the wheel graph...

The Gosset graph is distance-regular with diameter three. The induced subgraph of the neighborhood of any vertex in the Gosset graph is isomorphic to...

reversed edges of P2 from both paths. The remaining edges of P1 and P2 form a subgraph with two outgoing edges at s, two incoming edges at t, and one incoming...

 E), has a perfect matching if and only if for every subset U of V, the subgraph G − U has at most |U| odd components (connected components having an odd...

tree of the point set. The graph is a subgraph of the point set's Gabriel graph and therefore also a subgraph of its Delaunay triangulation. de Berg...

n {\displaystyle |V|=n} and | E | = m {\displaystyle |E|=m} is a tree subgraph of G {\displaystyle G} that contains all of its vertices and is of minimum...

graph G = (V,E), the set of triples of the vertex set V whose induced subgraph has an odd number of edges forms a two-graph on the set V. Every two-graph...

graph is polarized, that is, it decomposes into two entirely positive subgraphs that are joined by negative edges. In the interest of realism, a weaker...

bipartition, whose edges can be partitioned into n {\displaystyle n} induced subgraphs that are each matchings? What is the largest possible number of triples...

Proof without words that a hypercube graph is non-planar using Kuratowski's or Wagner's theorems and finding either K5 (top) or K3,3 (bottom) subgraphs...

hexagrams in I Ching that is also the basis for Kangxi radical 89 Yao graph, a subgraph that guarantees connectivity Yau (disambiguation) Yaw (disambiguation)...

from an empty edge set and then adding the next cheapest edge which is a subgraph of a complete tour. This greedy algorithm has been proven to yield at most...

neural circuit or simply a circuit) is a conceptual and computational subgraph within an artificial neural network, that performs a specific, interpretable...

adjacent only when the corresponding subtrees intersect. Thus, G forms a subgraph of the intersection graph of the subtrees. The full intersection graph...

two steps apart in G. The half-square or bipartite half is the induced subgraph of one side of the bipartition (say V) in the square graph: its vertex...

vertices Circuit of a matroid Circuit (neural network), a computational subgraph within an artificial neural network that performs a specific, interpretable...

polynomial time per triangulation. Connected subgraphs The connected subgraphs, and connected induced subgraphs, of a given connected graph form a state space...

pseudoforest subgraphs of G that have all the vertices of G. Such a pseudoforest need not have any edges, since for example the subgraph that has all...

fixed Turán graph as a subgraph. Via this theorem, similar bounds in extremal graph theory can be proven for any excluded subgraph, depending on the chromatic...

number of edges is at most a constant times the number of vertices in any subgraph), the maximum clique has bounded size and may be found exactly in linear...

saturated, because any complete type is isolated (implied) by the finite subgraph consisting of the variables and parameters used to define the type. Both...

matching is a subgraph of a half graph. If the chromatic number of a graph is uncountable, then the graph necessarily contains as a subgraph a half graph...

in the subgraph formed by the edges in X {\displaystyle X} and c {\displaystyle c} is the number of connected components of the same subgraph. The corank...

which a user specifies. This kind of search is achieved by looking for subgraph isomorphism (sometimes also called a monomorphism) and is a widely studied...

the occurrence of missing or spurious links in the network. Cliques are subgraphs in which every node is connected to every other node in the clique. As...

"chain graphs". In a chain graph, a connected component of the undirected subgraph is called a chain. Moralization adds an undirected edge between any two...

subgraph of vertices with at least degree k. This algorithm can only be applied to unweighted graphs. A minimum spanning tree is a tree-like subgraph...

called simply cycle cover) of a graph is a family of cycles which are subgraphs of G and contain all edges of G. If the cycles of the cover have no vertices...

called simply cycle cover) of a graph G is a set of cycles which are subgraphs of G and contain all vertices of G. If the cycles of the cover have no...

strong component algorithm Tarjan's strongly connected components algorithm Subgraph isomorphism problem Bitap algorithm: fuzzy algorithm that determines if...

specified size Linear spanning, a concept in abstract algebra Spanning tree, a subgraph which is a tree, containing all the vertices of a graph Søren Spanning...

greedy coloring algorithm with that ordering optimally colors every induced subgraph of the given graph. Perfectly orderable graphs form a special case of the...

most 3, and graphs of degree at most four that either contain a diamond subgraph or are not 4-edge-connected. In particular, the graphs that can be reduced...

pairs of leaves whose distance in T is at most k. That is, G is an induced subgraph of the graph power ⁠ T k {\displaystyle T^{k}} ⁠, induced by the leaves...

Data Analysis Heat Map Example: Subgraph of one of five hub nodes with a large degree of centrality in a genomic region in mice (Mus musculus) called the...

{\displaystyle X} of police locations to one of the connected components of the subgraph formed by deleting X {\displaystyle X} ; a robber can evade the police...

three distinct colors. Equivalently, in a star coloring, the induced subgraphs formed by the vertices of any two colors has connected components that...

graph G can be described as the forests F of G such that in the induced subgraph of V(G) − V(F), every connected component has a cycle. Since the flats...

are two of the few results about infinite graph coloring: If all finite subgraphs of an infinite graph G are k-colorable, then so is G, under the assumption...

trivial or a simple combination of the answers to the same problem on some subgraphs. Namely, for paths and polygons, the puzzle has no freedom; if the graph...

T.; Vesztergombi, K. (2008), "Convergent sequences of dense graphs. I. Subgraph frequencies, metric properties and testing", Advances in Mathematics, 219...

1 ( 1 ) | > 2 n − 1 {\displaystyle |g^{-1}(1)|>2^{n-1}} . Consider the subgraph G {\displaystyle G} of the hypercube (the graph on { 0 , 1 } n {\displaystyle...

relation to various parameters such as graph density, toughness, forbidden subgraphs and distance among other parameters. Dirac and Ore's theorems basically...

algorithm, an edge must be found that connects a vertex in a subgraph to a vertex outside the subgraph. Since P is connected, there will always be a path to...

degree of the two remaining vertices is 1. If a path graph occurs as a subgraph of another graph, it is a path in that graph. A planar graph is a graph...

{w^{2}+x^{2}+y^{2}+z^{2}}}={\sqrt {a+1-a}}=1} . We can also show that the subgraph K 3 , 3 {\displaystyle K_{3,3}} does not admit such a representation in...

undirected graph G is the smallest number d such that every non-empty subgraph of G has at least one vertex of degree at most d. If one repeatedly removes...

problem of exact matching of a graph to a part of another graph is called subgraph isomorphism problem. Inexact graph matching refers to matching problems...

Proof without words that a hypercube graph is non-planar using Kuratowski's or Wagner's theorems and finding either K5 (top) or K3,3 (bottom) subgraphs....

all k ≥ 2 {\displaystyle k\geq 2} , every k {\displaystyle k} -vertex subgraph has at most 2 k − 3 {\displaystyle 2k-3} edges, and such that the whole...

the vertices of the complete graph into the connected components of the subgraph formed by the given set of edges. In this way, the lattice of partitions...

from the formal definition of subtree used in graph theory, which is a subgraph that forms a tree – it need not include all descendants. For example, the...

L_{K}(E\setminus H)} , where E ∖ H {\displaystyle E\setminus H} is the subgraph of E {\displaystyle E} with vertex set ( E ∖ H ) 0 := E 0 ∖ H {\displaystyle...

With a Tangled Skein, a novel by Piers Anthony Skein (graph theory), a subgraph of a graph formed by paths connecting a given pair of vertices Skein (hash...

Erdős–Hajnal conjecture that in a family of graphs defined by an excluded induced subgraph, every graph has either a large clique or a large independent set. The...

{\displaystyle \Delta (G)\geq n/3} is class 2 if and only if it has an overfull subgraph S {\displaystyle S} satisfying Δ ( S ) = Δ ( G ) {\displaystyle \Delta...

set in a comparability graph corresponds to an antichain. Any induced subgraph of a comparability graph is itself a comparability graph, formed from the...

graphs and is "sandwiched" between two other graphs, one of which must be a subgraph and the other of which must be a supergraph of the desired graph. Graph...

number of any of its finite subgraphs; until the discovery of a family of 5-chromatic unit distance graphs in 2018, no subgraph of the infinite unit distance...

Glossary of graph theory § bridge; either an edge of a graph, a subgraph related to another subgraph, or a path related to a cycle, meeting a certain property...

being biologically relevant. The goal is to identify densely connected subgraphs within the graph, as these are likely to correspond to stable protein...

edges in a graph with n vertices not containing a subgraph isomorphic to H; see the Forbidden subgraph problem for more examples of problems involving the...

in 2010 Linkage (graph theory), the maximum min-degree of any of its subgraphs Linkage (horse), an American Thoroughbred racehorse Linkage (hierarchical...

itself distance-regular. And because it contains a hypercube as a spanning subgraph, it inherits from the hypercube all monotone graph properties, such as...

Bitfrost GrapheneOS Next-Generation Secure Computing Base OpenBSD Qubes Subgraph OS Tails Tinfoil Hat Linux Whonix BusKill USBKill Advanced Encryption Standard...

for any bipartite graph H, uniformly random graphons have the fewest subgraphs isomorphic to H when the edge density is fixed. He was awarded the Whitehead...

into the query, and then extract the data values associated with that subgraph. Data values can also be processed by functions, including aggregation...

Graphlets in mathematics are induced subgraph isomorphism classes in a graph, i.e. two graphlet occurrences are isomorphic, whereas two graphlets are non-isomorphic...

integral identity due to Godsil (1981). There is a similar relation for a subgraph G of Km,n and its complement in Km,n. This relation, due to Riordan (1958)...

April 2022). "HiddenCPG: Large-Scale Vulnerable Clone Detection Using Subgraph Isomorphism of Code Property Graphs". Proceedings of the ACM Web Conference...

undirected graph G is a chordal graph, on the same vertex set, that has G as a subgraph. A minimal chordal completion is a chordal completion such that any graph...

definition of claw-free graphs, graphs that do not have any claw as an induced subgraph. They are also one of the exceptional cases of the Whitney graph isomorphism...

acyclic coloring is a (proper) vertex coloring in which every 2-chromatic subgraph is acyclic. The acyclic chromatic number A(G) of a graph G is the fewest...

connected induced subgraph has the same distances as the whole graph). The gem shown is chordal but not distance-hereditary: in the subgraph induced by uvwx...

definition of perfect graphs via graph coloring, and provides a forbidden subgraph characterization of these graphs. Fomin, Fedor V.; Kratsch, Dieter; Novelli...

that the planar graphs are exactly those graphs that do not contain as a subgraph a subdivision of K5 or K3,3. Another result of his, also known as Wagner's...

The symmetries of a regular myriagon. Light blue lines show subgroups of index 2. The 5 boxed subgraphs are positionally related by index 5 subgroups....

and edges. G has no vertices of (finite) odd degree. Removing any finite subgraph S from G leaves at most two infinite connected components in the remaining...

of perfect graphs. An undirected graph is perfect if, in every induced subgraph, the chromatic number equals the size of the largest clique. In the comparability...

problem can be approximated by computing the minimum spanning tree of the subgraph of the metric closure of the graph induced by the terminal vertices, as...

Rubinstein, Aviad; Weinstein, Omri (2015), ETH hardness for densest-k-subgraph with perfect completeness, arXiv:1504.08352, Bibcode:2015arXiv150408352B...

graph K5. A graph is bowtie-free if it has no butterfly as an induced subgraph. The triangle-free graphs are bowtie-free graphs, since every butterfly...

sharp threshold unless it is correlated with the appearance of small subgraphs. This theorem has been widely applied to analyze random graphs and percolation...

2-trees. 2-connected series–parallel graphs are characterised by having no subgraph homeomorphic to K4. Series parallel graphs may also be characterized by...

graph theory, the Y-Δ transform means replacing a Y subgraph of a graph with the equivalent Δ subgraph. The transform preserves the number of edges in a...

Berge: given a k-connected graph G with independence number α(G), and any subgraph F of G with at most k − α(G) edges whose components are all paths, G has...

a generalization of Dijkstra's algorithm that reduces the size of the subgraph that must be explored, if additional information is available that provides...

two vertices which are endpoints of the matching edges (it is an induced subgraph). An induced matching can also be described as an independent set in the...

graph G has order dimension two if and only if the graph is a path or a subgraph of a path. For, when an incidence poset has order dimension two, its only...

metric, which says that two graphs are close if their distributions of subgraphs are close. The second is an edge discrepancy metric, which says two graphs...

36 simple maximal subgroups of order 168. These are the vertices of a subgraph, the U3(3) graph. A 168-subgroup has 14 maximal subgroups of order 24,...

Let G = ( V , E ) {\displaystyle G=(V,E)} be an arbitrary graph. If the subgraph G ′ = ( V , E ∖ X ) {\displaystyle G'=(V,E\setminus X)} is connected for...

every induced subgraph is well-colored) are exactly the cographs, the graphs that do not have a four-vertex path as an induced subgraph. Grundy, P. M...

of Kuratowski's theorem characterizing planar graphs by two forbidden subgraphs, one of which is K 3 , 3 {\displaystyle K_{3,3}} . The general question...

an arborescence. A multitree is a directed acyclic graph in which the subgraph reachable from any node forms a tree. Every polytree is a multitree. The...

quotient of G, in which the blocks are the connected components of the subgraph of G formed by the contracted edges. However, for quotients more generally...

maximum number of edges in a bipartite graph with no complete bipartite subgraph Ka,b. The Zarankiewicz crossing number conjecture in the mathematical field...

ascending subgraph decomposition, where a graph is decomposed into a sequence of subgraphs such that each of those graphs is isomorphic to a subgraph of the...

problems, including reachability and connectivity, shortest path, and subgraph detection. For many of these problems, the implied lower bounds have matching...

problem is known to be in NP, but it is unknown if it is NP-complete. Subgraph homeomorphism (for a fixed graph H) Graph genus Chordal graph completion...

Caen, D. (1983). "Extension of a theorem of Moon and Moser on complete subgraphs". Ars Combinatoria. 16: 5–10. Rödl, Vojtěch (1985). "On a packing and...

vertices (x, y). The strong components are the maximal strongly connected subgraphs. A connected rooted graph (or flow graph) is one where there exists a...

of edges in a bipartite graph that does not contain certain complete subgraphs. Another is the following so-called friendship theorem proved with Paul...

dominating set and the size of the smallest forbidden complete bipartite subgraph; that is, the problem is FPT on biclique-free graphs, a very general class...

that with e makes up a cutset, the minimal set of edges that connect two subgraphs. Versions of the Shannon switching game played on a directed graph and...

authorities; like HITS, SALSA also works on a focused subgraph which is topic-dependent. This focused subgraph is obtained by first finding a set of pages most...

equal to the sizes of the connected components of the vertex induced subgraphs. For a partition λ ⊢ n {\displaystyle \lambda \vdash n} , let z λ {\displaystyle...

subgraph characterization as the graphs that do not contain an induced cycle of length greater than three or an n-sun (n ≥ 3) as an induced subgraph....

their decompositions come from the action of a permutation on a fixed subgraph. They proved that for positive even integers m {\displaystyle m} and n...

two subsets, such that the induced subgraph formed by one of the two subsets is disconnected and the induced subgraph formed by the other subset is the...

|V_{n}(i)|=\textstyle {\binom {n}{i}}} . Let Q n ( i ) {\displaystyle Q_{n}(i)} be the subgraph of Q n {\displaystyle Q_{n}} induced by V n ( i ) ∪ V n ( i + 1 ) {\displaystyle...

β ≥ 1, the β-skeleton (with either definition) is a subgraph of the Gabriel graph, which is a subgraph of the Delaunay triangulation. If pq is an edge of...

its edges, and the removal of any two of its vertices leaves a connected subgraph. According to Steinitz's theorem, the icosahedral graph endowed with these...

vertices in a graph) should be low. Apart from this, the conductance of the subgraph induced by a cluster (called "internal conductance") can be used as well...

Eulerian subgraph of a graph G is a subgraph consisting of a subset of the edges of G, touching every vertex of G an even number of times. These subgraphs are...

An order embedding A → B is a graph isomorphism from A to an induced subgraph of B. (Category theoretically) A poset is a (small, thin, and skeletal)...

authorization format called XDI link contracts. Link contracts are XDI subgraphs that express the permissions that one XDI actor (person, organization...

vertices leaves a connected subgraph, and k-edge-connected if the removal of any (k − 1) edges leaves a connected subgraph. The following result is due...

index at most Δ(G) + 1. A k-degenerate graph is a graph in which each subgraph has a vertex of degree at most k. Indegree, outdegree for digraphs Degree...

regions can be used to prove that the Euclidean minimum spanning tree is a subgraph of other geometric graphs including the relative neighborhood graph and...

sets of vertices of a graph G {\displaystyle G} that induce complete subgraphs of G {\displaystyle G} . A clique complex forms a matroid if and only...

probability. The probabilities assigned to the posts can be arbitrary. A random subgraph of the bunkbed graph is then formed by independently deleting each edge...

A minimal pseudotriangulation is a pseudotriangulation T such that no subgraph of T is a pseudotriangulation covering the same convex region of the plane...

ray, in an infinite graph, is a semi-infinite path: a connected infinite subgraph in which one vertex has degree one and the rest have degree two. Halin...

with as few edges as possible that contains the given circle graph as a subgraph) may be found in O(n3) time. Tiskin (2010) has shown that a maximum clique...

nontrivial monotone graph property (a property that remains true for every subgraph of a graph with the property) requires a quadratic number of tests, but...

PMID 21177974. Berger, Bonnie (1992). "Tight Bounds for the Maximum Acyclic Subgraph Problem". Journal of Algorithms. 25: 1–18. doi:10.1006/jagm.1997.0864....

contains no complete subgraph on r vertices, in any green-red coloring of the edges of G there is either a green Kp or a red Kq subgraph. Some results are...

threshold graph if and only if it no four of its vertices form an induced subgraph that is a three-edge path graph, a four-edge cycle graph, or a two-edge...

notion of induced substructure is more restrictive than the notion of subgraph. For example, let G {\displaystyle G} be a graph consisting of two vertices...

the figure. The set of vertices reachable from starting vertex x0 form a subgraph with a shape resembling the Greek letter rho (ρ): a path of length μ from...

{\displaystyle i_{g}(W_{2k})=3k} . Subgraphs of Wheels : For k ≥ 13 {\displaystyle k\geq 13} , if G {\displaystyle G} is a subgraph of W k {\displaystyle W_{k}}...

in which Tutte provided gadgets for reducing the problem of finding a subgraph with given degree constraints to a perfect matching problem. However, the...

subset of the other. Equivalently, the Desargues graph is the induced subgraph of the 5-dimensional hypercube determined by the vertices of weight 2 and...

2-trees. This means that their maximum degree is three and that they are subgraphs of series-parallel graphs. All other graphs have carving width at least...

analysis. Han et al. propose the creation of a disambiguation graph (a subgraph of the knowledge base which contains candidate entities). This graph is...

possible number of triangles, the largest possible number of tetrahedral subgraphs, the largest possible number of cliques, and the largest possible number...

Barman, S. (2015). "Approximating Nash equilibria and dense bipartite subgraphs via an approximate version of Carathéodory's theorem". Proc. 47th Annu...

equitable coloring is that it is an embedding of the given graph as a subgraph of a Turán graph with the same set of vertices. There are two kinds of...

argument is generalized to considering configurations, which are connected subgraphs of G with the degree of each vertex (in G) specified. For example, the...

number w such that there exists an interval graph H containing G as a subgraph, with the largest clique in H having w + 1 vertices. For trees (viewed...

T} , the graphs that do not contain T {\displaystyle T} as an induced subgraph are χ {\displaystyle \chi } -bounded. For instance, this would include...

no induced subgraph isomorphic to a path with four vertices, the clique-width of many graph classes defined by forbidden induced subgraphs has been classified...

the vertex and edge functions appearing in that set. Furthermore, the subgraphs of a multigraph G are in bijection with the graph homomorphisms from G...

wolfram.com. Retrieved 7 February 2025. A clique of a graph G is a complete subgraph of G, and the clique of largest possible size is referred to as a maximum...

hence a bipartite permutation graph) if and only if it contains no induced subgraph isomorphic to certain forbidden configurations. Every biconvex graph is...

others (if any) have degree 2. Paths are often important in their role as subgraphs of other graphs, in which case they are called paths in that graph. A...

Hedrlín, Z.; Mendelsohn, E. (1969). "The Category of Graphs with a Given Subgraph-with Applications to Topology and Algebra". Canadian Journal of Mathematics...

(one graph is an amalgamation of another). Similar relationships include subgraphs and minors. Amalgamations can provide a way to reduce a graph to a simpler...

The Gabriel Graph, a subgraph of the Delaunay triangulation in which the circle surrounding each edge does not enclose any other points of the graph...

other). A claw-free interval graph. A graph that does not have an induced subgraph isomorphic to a claw K 1 , 3 {\displaystyle K_{1,3}} , net (a triangle...

sampling step which partitions a graph into two subgraphs by randomly selecting edges to include in each subgraph. The algorithm recursively finds the minimum...

edges among U {\displaystyle U} (equivalently, the number of edges in the subgraph induced by the vertex set U {\displaystyle U} ). It can be shown that the...

network that contains a spanning polyhedral subgraph: first, construct a polyhedron with a midsphere for this subgraph. Then, to handle a message that should...

undirected graph, the maximum number of shared vertices of any pair of subgraphs formed by removing an edge from the tree. Clique-width of a graph, the...

sorting Feedback arc set, a set of edges whose removal allows the remaining subgraph to be topologically sorted Tarjan's strongly connected components algorithm...

perfect matching is a spanning 1-regular subgraph, a.k.a. a 1-factor. In general, a spanning k-regular subgraph is a k-factor. A spectral characterization...

bond thresholds have been given by Norrenbrock. The Gabriel graph is a subgraph of the Delaunay triangulation. It can be found in linear time if the Delaunay...

can have any cardinality, there are several notions of the concept of a subgraph, called subhypergraphs, partial hypergraphs and section hypergraphs. Let...

approach to the automatic indexing of organic reactions using Maximum common subgraph isomorphism algorithms, which became foundational for many reaction database...

the property of being dually chordal is not hereditary, i.e., induced subgraphs of a dually chordal graph are not necessarily dually chordal (hereditarily...

math.ndsu.nodak.edu. Retrieved 2019-12-21. Huang, Hao (2019). "Induced subgraphs of hypercubes and a proof of the Sensitivity Conjecture". Annals of Mathematics...

branches, or orthogonal mirrors) requires at least one active node in each subgraph. All regular polytopes, represented by Schläfli symbol {p, q, r, ...},...

Blossom (functional), a functional for polynomials Blossom (graph theory), a subgraph in which removing any vertex leaves a graph with a perfect matching HMS...

of highly central nodes in a graph does not make an internally cohesive subgraph (Borgatti & Everett, 2000)... The concept was first introduced into economics...

Erdős–Gallai theorem From Shahriari (2022, p. 48): "Definition 2.17 (Graphs & Subgraphs). A simple graph (or just a graph) G is a pair of sets (V, E) where V...

1-planar graphs are K6, K1,1,1,6, K1,1,2,3, K2,2,2,2, K1,1,1,2,2, and their subgraphs. The minimal non-1-planar complete multipartite graphs are K3,7, K4,5...

smallest number of r {\displaystyle r} -edges such that every induced subgraph on k {\displaystyle k} vertices contains an edge. This number was determined...

the graph is planar, or an obstacle to planarity such as a Kuratowski subgraph if it is not. Planarity testing algorithms typically take advantage of...

lens-based beta skeleton. It is a subgraph of the Delaunay triangulation. In turn, the Euclidean minimum spanning tree is a subgraph of it, from which it follows...

speaking, F {\displaystyle F} is a common graph if it "commonly" appears as a subgraph, in a sense that the total number of copies of F {\displaystyle F} in any...

Cayley configuration spaces below. Closure under subgraphs. Flattenability is closed under taking subgraphs. To see this, observe that for some graph G {\displaystyle...

which are not symmetric can be formed as subgraphs of these complete bi-partite graphs in certain cases. Subgraphs of complete bipartite graphs Km,n exist...

path in a hypercube; it can be viewed as a special case of the induced subgraph isomorphism problem. There is a similar problem of finding long induced...

{\displaystyle \gamma (H)=\gamma _{\text{cer}}(H)} for every induced connected subgraph H ≠ K 2 {\displaystyle H\neq K_{2}} of G {\displaystyle G} . A graph is...

into the social sciences, and coining the term "clique" for a complete subgraph in graph theory. In 1966, Luce was elected to the American Academy of Arts...

d-vertex-connected: the removal of any d − 1 vertices leaves a connected subgraph. For instance, for a three-dimensional polyhedron, even if two of its vertices...

{Q} } ). Mathematics portal Connected component (graph theory) – Maximal subgraph whose vertices can reach each otherPages displaying short descriptions...

where t ( G ) {\displaystyle t(G)} is the number of triangles (3-cycle subgraphs) in G {\displaystyle G} . The coefficient of x 1 {\displaystyle x^{1}}...

u and v. The vertices in a bag can be thought of as the terminals of a subgraph of G, represented by the subtree of the tree decomposition descending from...

estimated gross profits k-factor (graph theory), a spanning k-regular subgraph in graph theory K-factor, the circular segment of earth profile that blocks...

smallest graph which contains every member of a given family of graphs as subgraphs. In a series of works with Paul Erdős, Chung determined the sizes and...

substructures. Furthermore, there exist also approaches using maximum common subgraph searches or graph kernels. Typically QSAR models derived from non linear...

most one cycle in each of its connected components. Any vertex-induced subgraph with more edges than vertices corresponds to a set of keys for which there...

from u {\displaystyle u} to v {\displaystyle v} in the subgraph. At each step down the subgraph there are two choices one can make to form such a path...

{\displaystyle |V_{i}|=n_{i}} , and V i {\displaystyle V_{i}} spans a connected subgraph. In 1987 the necklace splitting problem was solved by Noga Alon using the...

graph is colored with two colors, there will always exist a solid color subgraph of each color. Other contributors to the study of unavoidable patterns...

one heavy edge from each non-leaf node to one of its children, then the subgraph formed by the heavy edges consists of a set of paths, with each non-leaf...

cycles. A given graph has a Hamiltonian decomposition if and only if the subgraph formed by removing an arbitrary vertex from the graph has linear arboricity...

edges of G and the feasible sets be the edge set of each forest (i.e. subgraph containing no cycle) of G. This set system is called the cycle matroid...

(1979), "Minimal decompositions of two graphs into pairwise isomorphic subgraphs", Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph...

graph coloring; Hamiltonian cycles and Euler tours; extremal graph theory; subgraph counting problems including connections to permutations, derangements,...

different colours, then for some i between 1 and c, it must contain a complete subgraph of order ni whose edges are all colour i. The special case above has c...

k))\leq n.} Each Johnson graph is locally grid, meaning that the induced subgraph of the neighbors of any vertex is a rook's graph. More precisely, in the...

graph that has no Kt,t (complete bipartite graph with 2t vertices) as a subgraph. A family of graphs is biclique-free if there exists a number t such that...

characterize the organizational architecture of Complex networks, such as the "subgraph centrality", "communicability", "spectral scaling", among others. His generalization...

S2CID 16082154. Chiba, Norishige; Nishizeki, Takao (1985), "Arboricity and subgraph listing algorithms", SIAM Journal on Computing, 14 (1): 210–223, doi:10...

mixed graph, the ring star problem aims to find a minimum cost ring star subgraph formed by a cycle (ring part) and a set of arcs (star part) such that each...

Nešetřil, Jaroslav; Poljak, Svatopluk (1985). "On the complexity of the subgraph problem". Commentationes Mathematicae Universitatis Carolinae. 26 (2):...

cannot be decomposed via nesting or sequencing using a chosen pattern of subgraphs, for example the primitives of structured programming. Theoretical research...

published in 1993 by Brouwer, Dejter and Thomassen as a self-complementary subgraph of the Dejter graph. In 1972, Bouwer was already talking of a 112-vertices...

The degeneracy of a graph G is the smallest number d such that every subgraph of G has a vertex with degree d or less. Every graph has a degeneracy ordering...

a dominating set S {\displaystyle S} of vertices such that the induced subgraph G [ S ] {\displaystyle G[S]} contains at least one perfect matching. The...

5-vertex cycle graph. This means that these two subgraphs are connected by all possible edges from one subgraph to the other. The resulting graph has 11 vertices...

archived from the original on 2013-12-24, retrieved 2013-12-24. "GitHub – subgraph/Orchid". 7 March 2019. Archived from the original on 9 August 2017. Retrieved...

embeddings: They are the median graphs that do not contain as an induced subgraph any member of an infinite family of forbidden graphs. These forbidden graphs...

combinatorial embedding is produced as a witness. If not, a Kuratowski subgraph is returned. These values can then be passed directly to checker functions...

view the product itself as rooted, at (g1, h1). The rooted product is a subgraph of the cartesian product of the same two graphs. The rooted product is...

that it is optimal both for the graph itself and for all of its induced subgraphs. The perfectly orderable graphs (which include chordal graphs, comparability...

S2CID 119705062. Grötschel, M.; Jünger, M.; Reinelt, G. (1985a). "On the acyclic subgraph polytope". Mathematical Programming. 33: 28–42. doi:10.1007/BF01582009...

tree-depth ordered by the induced subgraph relation form a well-quasi-order, as do the cographs ordered by induced subgraphs. Let X 1 {\displaystyle X_{1}}...

"Beating the Random Ordering is Hard: Inapproximability of Maximum Acyclic Subgraph". 2008 49th Annual IEEE Symposium on Foundations of Computer Science. pp...

G, there exists a highly irregular graph H containing G as an induced subgraph. This last observation is analogous to a result of Dénes Kőnig, which states...

During the preprocessing phase the graph is partitioned into disjointed subgraphs S i , 0 ≤ i ≤ K {\displaystyle S_{i},\,0\leq i\leq K} with small diameter...

brain graphs form a connected subgraph around the brainstem. By allowing gradually less frequent edges, this core subgraph grows continuously, as a shrub...

ω(H is the number of components of H and the maximum is taken over all subgraphs H for which the denominator is not zero. They also defined the strength...

into two subgraphs with approximately equal numbers of vertices. After finding an approximate cut, their algorithm arranges the two subgraphs on each side...

any two vertices leaves a subgraph the edges of which can be colored with only three colors. A 3-coloring of this subgraph can be simply described: after...

property of a graph which also holds for (is "inherited" by) its induced subgraphs. Equivalently, a hereditary property is preserved by the removal of vertices...

combinatorics". Let t(H, G) = ⁠# labeled copies of H in G/v(G)v(H)⁠, known as the subgraph density (in particular, t(K2, G) is the edge density of G). A sequence...

graph spanners has been known in graph theory: t-spanners are spanning subgraphs of graphs with similar dilation property, where distances between graph...

which states that every graph is uniquely determined by the ensemble of subgraphs formed by deleting one vertex in each possible way. He also proved a special...

removing S from the graph, partitions the graph into two smaller connected subgraphs A and B, each of which has at most n⁄2 vertices. If r ≤ c (as in the illustration)...

bounded expansion, the subgraph isomorphism problem, of finding whether a pattern graph of bounded size exists as a subgraph of a larger graph, can be...

or maximal complete subgraph in the complementary graph. A maximal clique is a set of vertices that induces a complete subgraph, and that is not a subset...

is a sequence of its vertices with the property that, for any induced subgraph of G, a greedy coloring algorithm that colors the vertices in the induced...

(d+1)} -dimensional polytope on R d {\displaystyle \mathbb {R} ^{d}} . The subgraph induced by these triangulations in the flip graph of A {\displaystyle {\mathcal...

2019-11-28. Retrieved 2019-11-29. Rossman, Benjamin (2019). "Lower Bounds for Subgraph Isomorphism". In Boyan, Sirakov; De Souza, Paulo Ney; Viana, Marcelo (eds...

characterization of the weakly bipartite graphs (graphs whose bipartite subgraph polytope is 0-1). Satoru Iwata, Lisa Fleischer, Satoru Fujishige, and Alexander...

In a control-flow graph (CFG), a SESE region is typically defined as a subgraph with: One entry edge that dominates all nodes in the region; One exit edge...

that is, its density can be factorized over the cliques (or complete subgraphs) of the graph. The relationship between Markov and Gibbs random fields...

work with Chudnovsky, and obtained several more results about induced subgraphs, in particular (with Cornuéjols, Liu, and Vušković) a polynomial-time...

lines tangent to the convex hull of the points. This graph contains as a subgraph the thrackle of diameter pairs. The diameters of the Reinhardt polygons...

few earlier neighbors, it has low chromatic number. When every finite subgraph has an ordering of this type in which the number of previous neighbors...

denotes the square of a graph and the square brackets denote an induced subgraph. For instance, the bipartite half of the complete bipartite graph Kn,n...

larger maximal planar graph in which G {\displaystyle G} is an induced subgraph. Constructing a circle packing from this larger graph, and then removing...

during delete operations. For each i between 0 and L, define Gi as the subgraph consisting of edges that are at level i or less, and Fi a spanning forest...

description) In terms of graph theory, any syntactic tree or connected subgraph of a tree is a catena. Any individual element (word or morph) or combination...

contains B p {\displaystyle B_{p}} as a subgraph, or its complement graph contains B q {\displaystyle B_{q}} as a subgraph. If 1 ≤ q {\displaystyle 1\leq q}...

Babai, László; Sós, Vera T. (1985). "Sidon sets in groups and induced subgraphs of Cayley graphs" (PDF). European Journal of Combinatorics. 6 (2): 101–114...

{\displaystyle \{t,t,s\}=\{t,s\}} , trivializing the permutation of the two subgraphs of shape t {\displaystyle t} ). This graph model enables an implementation...

learning, and showed that it contained the minimum spanning tree, and was a subgraph of the Delaunay triangulation. Three other well known proximity graphs...

and more, can be represented as graphs, which include a wide variety of subgraphs. One important local property of networks are so-called network motifs...

partition into a sequence of subgraphs called "ears", in which the first subgraph in the sequence is a cycle and each subsequent subgraph is a path, with the two...

element. The intersection number equals the smallest number of cliques (subgraphs with edges between all pairs of vertices) needed to cover all of the edges...

C^{*}(E\setminus H)} , where E ∖ H {\displaystyle E\setminus H} is the subgraph of E {\displaystyle E} with vertex set ( E ∖ H ) 0 := E 0 ∖ H {\displaystyle...

without backtracking. Since the graph of the instance they produce is a subgraph of the induced graph, if the induced width is bounded by a constant the...

graphs, 3-connected 3-regular bipartite graphs, subgraphs of the square grid graph, cubic subgraphs of the square grid graph. However, for some special...

whenever two lines meet. This graph was analyzed in the 19th century using subgraphs such as the Schläfli double six configuration. The complementary graph...

G {\displaystyle G} is AT-free if and only if every connected induced subgraph H {\displaystyle H} satisfies the spine property: for every nonadjacent...

Ajtai, M.; Chvátal, V.; Newborn, M.; Szemerédi, E. (1982), "Crossing-free subgraphs", Theory and Practice of Combinatorics, North-Holland Mathematics Studies...

{\textstyle (x,v=u\cdot Mg(x))} , one produces a uniform simulation over the subgraph of M g ( x ) {\textstyle Mg(x)} . Accepting only pairs such that u < f...

of all copies of the original variables is usually formulated as: the subgraph induced by the nodes associated to an original variable is connected. A...

"Inapproximability of maximum biclique problems, minimum k-cut and densest at-least-k-subgraph from the small set expansion hypothesis", Algorithms, 11 (1): P10:1–P10:22...

forms a connected subgraph. Symmetrically, if S is connected, then the edges dual to the complement of S form an acyclic subgraph. Therefore, when S...

two mutually inscribed quadrilaterals, has the Möbius–Kantor graph, a subgraph of Q4, as its Levi graph. Al-Dhahir, M. W. (1956), "A class of configurations...

their work on color coding, an application of the probabilistic method to subgraph isomorphism.[A] His work with Zwick on sparse matrix multiplication received...

a graph that is not 3-connected may result in degeneracies, in which subgraphs of the given graph collapse onto a point or a line segment; however, an...

Marc; Tejel, Javier (2000), "Lower bounds on the number of crossing-free subgraphs of K N {\displaystyle K_{N}} ", Computational Geometry: Theory & Applications...

Kazuo; Tamaki, Hisao; Tokuyama, Takeshi (2000), "Greedily finding a dense subgraph", Journal of Algorithms, 34 (2): 203–221, doi:10.1006/jagm.1999.1062, MR 1734799...

Graph nodes linked to, but not part of, a subgraph...

{\displaystyle f} be the fractional matching. Let H {\displaystyle H} be a subgraph of G {\displaystyle G} containing only the edges e {\displaystyle e} with...

Fortune, Steven; Hopcroft, John; Wyllie, James (1980-02-01). "The directed subgraph homeomorphism problem". Theoretical Computer Science. 10 (2): 111–121....

bar-joint frameworks if and only if G {\displaystyle G} has a spanning subgraph G ′ = ( V , E ′ ) {\displaystyle G'=(V,E')} such that | E ′ | = 2 | V |...

 19–37. doi:10.1002/cpa.3160370104 with Noga Alon and Gil Kalai: "Regular subgraphs of almost regular graphs", Journal of Combinatorial Theory, Series B,...

directed acyclic graph G has an upward planar drawing if and only if G is a subgraph of an st-planar graph. It is possible to find an st-numbering, and a bipolar...

is well-colored. A graph is hereditarily well-colored if every induced subgraph is well-colored. The hereditarily well-colored graphs are exactly the cographs...

first converted to internal graph representations which are searched for subgraph isomorphism. SMIRKS, a superset of "reaction SMILES" and a subset of "reaction...

flow. Ford–Fulkerson algorithm Maximum flow problem This means that the subgraph resulting from removing all saturated edges (edges ( u , v ) {\displaystyle...

That is, λ G ( v ) {\displaystyle \lambda _{G}(v)} is the number of subgraphs of G {\displaystyle G} with 3 edges and 3 vertices, one of which is v...

graph in which every subgraph is perfect. The deletion of any vertex from a minimal imperfect graph leaves a uniquely colorable subgraph. A uniquely edge-colorable...

nodes and power-law exponent γ > 3 {\displaystyle \gamma >3} , the induced subgraph constructed by vertices with degrees larger than log ⁡ n × log ∗ ⁡ n {\displaystyle...

graph counting lemma that estimates number of copies of a fixed graph as a subgraph of a larger graph. There are several distinct formulations of the method...

in a graph G form an independent set, or more generally if the induced subgraph for this set of vertices is a forest, then G must be of class one. Erdős...

and the nut graphs, singular graphs all of whose nontrivial induced subgraphs are non-singular. She is a professor of mathematics at the University...

Dowling lattice by excluding all partial partitions such that the induced subgraph on some Bi is disconnected. The characteristic polynomial of this matroid...

expect a p | E ( H ) | {\displaystyle p^{|E(H)|}} fraction of possible subgraphs to be a copy of H {\displaystyle H} if each edge exists with probability...

list/matrix storage, and graph databases (e.g., Neo4j). Query Types: Subgraphs, patterns, or textual queries. Applications: Social network analysis....

partitioned into an independent set and a small number of complete bipartite subgraphs, inferring business relationships among autonomous subsystems of the internet...