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Embedding a graph in a topological space, often Euclidean
In topological graph theory, an embedding (also spelled imbedding) of a graph G {\displaystyle G} on a surface Σ {\displaystyle \Sigma } is a representation
Graph_embedding
Dimensionality reduction of graph-based semantic data objects [machine learning task]
In representation learning, knowledge graph embedding (KGE), also called knowledge representation learning (KRL), or multi-relation learning, is a machine
Knowledge_graph_embedding
Graph that can be embedded in the plane
Such a drawing is called a plane graph, or a planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node
Planar_graph
Type of knowledge base
such as data reasoning, node embedding, and ontology development on knowledge bases. In contrast, virtual knowledge graphs do not store information in
Knowledge_graph
Graph layout on multiple half-planes
In graph theory, a book embedding is a generalization of planar embedding of a graph to embeddings in a book, a collection of half-planes all having the
Book_embedding
Branch of the mathematical field of graph theory
topological graph theory is a branch of graph theory. It studies the embedding of graphs in surfaces, spatial embeddings of graphs, and graphs as topological
Topological_graph_theory
Graph representing faces of another graph
of embedding of the graph G, so it is a property of plane graphs (graphs that are already embedded in the plane) rather than planar graphs (graphs that
Dual_graph
Embedding a graph in 3D space with no cycles interlinked
Euclidean space in such a way that no two cycles of the graph are linked. A flat embedding is an embedding with the property that every cycle is the boundary
Linkless_embedding
Cycles in a graph that cover each edge twice
bridgeless graph has a nowhere-zero 5-flow. A stronger type of embedding than a circular embedding is a polyhedral embedding, an embedding of a graph on a surface
Cycle_double_cover
Planar graph drawn by relaxing springs
In graph drawing and geometric graph theory, a Tutte embedding or barycentric embedding of a simple, 3-vertex-connected, planar graph is a crossing-free
Tutte_embedding
Cubic graph with 10 vertices and 15 edges
This is the embedding given by the hemi-dodecahedron construction of the Petersen graph (shown in the figure). The projective plane embedding can also be
Petersen_graph
Area of discrete mathematics
imbedding) of a graph in surface and linkless embedding, graph minors, crossing number, map coloring, and voltage graph. The embedding of a graph in a surface
Graph_theory
Representation learning technique
becomes less reliable for large embedding vectors. Latent space Feature extraction Dimensionality reduction Word embedding Neural network Reinforcement learning
Embedding_(machine_learning)
A planar graph is a graph that has such an embedding onto the Euclidean plane, and a toroidal graph is a graph that has such an embedding onto a torus
Glossary_of_graph_theory
Graph with at most one crossing per edge
1-planar graph, one of the most natural generalizations of planar graphs, is drawn that way, the drawing is called a 1-plane graph or 1-planar embedding of
1-planar_graph
Methodic assignment of colors to elements of a graph
In graph theory, graph coloring is a methodic assignment of labels traditionally called "colors" to elements of a graph. The assignment is subject to certain
Graph_coloring
Graph of numbers differing by a square
finds embeddings of the Paley graphs of order q ≡ 1 (mod 8) that are highly symmetric and self-dual, generalizing a natural embedding of the Paley graph of
Paley_graph
Graph often embedded in the Klein bottle
six colors are sometimes necessary in this case. This embedding is the Petrie dual of its embedding in the projective plane shown below. It is Hamiltonian
Franklin_graph
Planar graph embedding where edges map to straight-line segments
geometric graph theory, a planar straight-line graph (PSLG), also called a straight-line plane graph or plane straight-line graph, is an embedding of a planar
Planar_straight-line_graph
Graph with all vertices of degree 3
of graph theory, a cubic graph is a graph in which all vertices have degree three. In other words, a cubic graph is a 3-regular graph. Cubic graphs are
Cubic_graph
Mathematical puzzle of avoiding crossings
a graph embedding in the plane. The impossibility of the puzzle corresponds to the fact that K 3 , 3 {\displaystyle K_{3,3}} is not a planar graph. Multiple
Three_utilities_problem
Visual technique in topological graph theory
from the graph, allowing holes through which the rest of the embedding can be seen. Ribbon graphs are also called fat graphs. In a ribbon graph representation
Ribbon_graph
Algorithmic problem of finding non-crossing drawings
the output of a planarity testing algorithm may be a planar graph embedding, if the graph is planar, or an obstacle to planarity such as a Kuratowski
Planarity_testing
Graph with edges of length one, able to be drawn without crossings
That is, it is a graph that has an embedding which is simultaneously a unit distance graph and a plane graph. Informally, matchstick graphs can be made by
Matchstick_graph
Non-crossing graph with vertices on outer face
outerplanarity. A 1-outerplanar embedding of a graph is the same as an outerplanar embedding. For k > 1 a planar embedding is said to be k-outerplanar if
Outerplanar_graph
placement provides a simultaneous embedding. There are two restricted models: simultaneous geometric embedding, where each graph must be drawn planarly with
Simultaneous_embedding
Problem in network theory
based methods. Graph embeddings also offer a convenient way to predict links. Graph embedding algorithms, such as Node2vec, learn an embedding space in which
Link_prediction
Undirected, connected, and acyclic graph
to an embedding of the tree in the plane, with the root at the top and the children of each vertex lower than that vertex. Given an embedding of a rooted
Tree_(graph_theory)
Graph in which every two vertices are adjacent
any three-dimensional embedding of K7 contains a Hamiltonian cycle that is embedded in space as a nontrivial knot. Complete graphs on n {\displaystyle n}
Complete_graph
Symmetric bipartite cubic graph with 16 vertices and 24 edges
seen in the diagram above. Genus 2 embedding Genus 3 embedding The automorphism group of the Möbius–Kantor graph is a group of order 96. It acts transitively
Möbius–Kantor_graph
Topics referred to by the same term
contained within another instance Graph embedding, in topological graph theory Embedded generation, of energy Embedding, a part of sample preparation for
Embedded
In distributed computing and geometric graph theory, greedy embedding is a process of assigning coordinates to the nodes of a telecommunications network
Greedy_embedding
Topics referred to by the same term
sequence Geometric genus In graph embedding, the genus of the graph is the genus of the surface in which it can be embedded In the theory of numerical
Genus_(disambiguation)
Operation combining two oriented knots
the planar graphs is provided by the graphs with linkless embeddings and knotless embeddings. A linkless embedding is an embedding of the graph with the
Knot_(mathematics)
Visualization of node-link graphs
a graph drawing represents a graph embedding. However, nonplanar graphs frequently arise in applications, so graph drawing algorithms must generally allow
Graph_drawing
Type of monotone function
must be an order embedding. However, not every order embedding is a coretraction. As a trivial example, the unique order embedding f : ∅ → { 1 } {\displaystyle
Order_embedding
Geometric graph with unit edge lengths
In mathematics, particularly geometric graph theory, a unit distance graph is a graph formed from a collection of points in the Euclidean plane by connecting
Unit_distance_graph
Bipartite 4-regular graph with 20 nodes and 40 edges
complete graph. Beyond the investigation of its symmetry, it has also been investigated as a counterexample for certain questions of graph embedding. Semi-symmetric
Folkman_graph
Text-structure representation using graph models
etc. Graph-based methods for NLP and Semantic Web Representation learning methods for knowledge graphs (i.e., knowledge graph embedding) Using graphs-based
Text_graph
Undirected cubic graph with 12 vertices and 18 edges
the Möbius strip itself) form an embedding of Tietze's graph. Tietze's graph may be formed from the Petersen graph by replacing one of its vertices with
Tietze's_graph
Physical simulation to visualize graphs
Koren, Yehuda (2002), "Graph drawing by high-dimensional embedding", Proceedings of the 9th International Symposium on Graph Drawing, Springer, pp. 207–219
Force-directed_graph_drawing
Graph representing edges of another graph
a planar graph G has maximum vertex degree three, its line graph is planar, and every planar embedding of G can be extended to an embedding of L(G). However
Line_graph
Mathematical result
the embedding is a random orthogonal projection. The lemma has applications in compressed sensing, manifold learning, dimensionality reduction, graph embedding
Johnson–Lindenstrauss_lemma
Matrix representation of a graph
low-dimensional embeddings that appear in many machine learning applications and determines a spectral layout in graph drawing. Graph-based signal processing
Laplacian_matrix
Symmetric tessellation of a closed surface
lines. Topological graph theory Abstract polytope Planar graph Toroidal graph Graph embedding Regular tiling Platonic solid Platonic graph Nedela (2007) Coxeter
Regular_map_(graph_theory)
Subgraph with contracted edges
contraction of edges can increase the genus of the embedding; therefore, planar graphs and the graphs embeddable on any fixed surface form minor-closed families
Graph_minor
Graph whose embedding in a Euclidean space forms a regular tiling
In graph theory, a lattice graph, mesh graph, or grid graph is a graph whose drawing, embedded in some Euclidean space R n {\displaystyle \mathbb {R}
Lattice_graph
Graph of chess rook moves
In graph theory, a rook's graph is an undirected graph that represents all legal moves of the rook chess piece on a chessboard. Each vertex of a rook's
Rook's_graph
Graph able to be embedded on a torus
the mathematical field of graph theory, a toroidal graph is a graph that can be embedded on a torus. In other words, the graph's vertices and edges can be
Toroidal_graph
On graph drawing with integer edge lengths
planar graph have an integral Fáry embedding? More unsolved problems in mathematics In mathematics, Harborth's conjecture states that every planar graph has
Harborth's_conjecture
24-vertex symmetric bipartite cubic graph
the Nauru graph. The dual graph of this embedding is a symmetric 6-regular graph with 12 vertices and 36 edges. The other symmetric embedding of the Nauru
Nauru_graph
Subset of artificial intelligence
Insurance Internet fraud detection Investment management Knowledge graph embedding Linguistics Machine learning control Machine perception Machine translation
Machine_learning
Projection of data onto lower-dimensional manifolds
linear embedding, spectral embedding, local tangent space alignment, multidimensional scaling, and t-distributed stochastic neighbor embedding. Waffles
Nonlinear dimensionality reduction
Nonlinear_dimensionality_reduction
Fast-growing function
homeomorphically embeddable into (i.e. is a graph minor of) G j {\displaystyle G_{j}} . The Robertson–Seymour theorem proves that subcubic graphs (simple or
Friedman's_SSCG_function
Set of learning techniques in machine learning
Chang, Kevin Chen-Chuan (September 2018). "A Comprehensive Survey of Graph Embedding: Problems, Techniques, and Applications". IEEE Transactions on Knowledge
Feature_learning
Cycle graph plus universal vertex
wheel graph with vertex set {1, 2, …, v} in which the vertex 1 is a universal vertex. Wheel graphs are planar graphs, and have a unique planar embedding. More
Wheel_graph
Graph drawing used to study Riemann surfaces
to exist, the graph must be bipartite. The faces of the embedding are required to be topological disks. The surface and the embedding may be described
Dessin_d'enfant
Family of graphs with 2n nodes and n(n-1) edges
crown graph may be embedded into four-dimensional Euclidean space in such a way that all of its edges have unit length. However, this embedding may also
Crown_graph
Format for expressing RDF statements in HTML documents
for embedding rich metadata within web documents. The Resource Description Framework (RDF) data-model mapping enables the use of RDFs for embedding RDF
RDFa
Mathematical tree with cycle through leaves
embedding), and the cycle connects the leaves in their clockwise ordering in this embedding. Thus, the cycle forms the outer face of the Halin graph,
Halin_graph
Graph representing connectivity between cliques of another graph
simplex graph of a complete graph is a hypercube graph, and the simplex graph of a cycle graph of length four or more is a gear graph. The simplex graph of
Simplex_graph
Puzzle computer game involving planar graphs
to eliminate all of the crossings and construct a straight-line embedding of the graph by moving the vertices one by one into better positions. The game
Planarity
A Euclidean graph (a graph embedded in some Euclidean space) is periodic if there exists a basis of that Euclidean space whose corresponding translations
Periodic_graph_(geometry)
Planar, undirected graph with 2n vertices and 3n-2 edges
mathematical field of graph theory, the ladder graph Ln is a planar, undirected graph with 2n vertices and 3n − 2 edges. The ladder graph can be obtained as
Ladder_graph
Undirected graph named after S. S. Shrikhande
vertex is surrounded by six triangles. Thus, the Shrikhande graph is a toroidal graph. The embedding forms a regular map in the torus, with 32 triangular faces
Shrikhande_graph
Largest independent set of paired elements
possible. The optimal embedding can then be obtained by pairing edges within each component and inserting each pair into an embedding, one pair at a time
Matroid_parity_problem
On coloring the edges of graphs
polyhedral embedding is a graph embedding such that every face of the embedding is topologically a disk and such that the dual graph of the embedding is simple
Vizing's_theorem
Type of planar graph
In graph theory, a k-outerplanar graph is a planar graph that has a planar embedding in which the vertices belong to at most k {\displaystyle k} concentric
K-outerplanar_graph
topology and graph theory, a map is a subdivision of a surface such as the Euclidean plane into interior-disjoint regions, formed by embedding a graph onto the
Map_(graph_theory)
Undirected unit-distance graph requiring four colors
who constructed it (with a non-planar embedding) as a unit distance graph that requires four colors in any graph coloring. Thus, like the simpler Moser
Golomb_graph
Number of planar subgraphs to cover a graph
of simultaneous embedding. If two or more planar graphs all share the same vertex set, then it is possible to embed all these graphs in the plane, with
Thickness_(graph_theory)
Graph property
Colin de Verdière's invariant is a graph parameter μ ( G ) {\displaystyle \mu (G)} for any graph G, introduced by Yves Colin de Verdière in 1990. It was
Colin de Verdière graph invariant
Colin_de_Verdière_graph_invariant
Representation of a graph's triconnected components
planar graph is 3-connected, it has a unique planar embedding up to the choice of which face is the outer face and of orientation of the embedding: the
SPQR_tree
Planar graph with 23 vertices and 63 edges
mathematical field of graph theory, the Kittell graph is a planar graph with 23 vertices and 63 edges. Its unique planar embedding has 42 triangular faces
Kittell_graph
Characterization of planar graphs by matroids
way that every face of the embedding is a topological disk, then the dual graph of the embedding is defined as the graph (or in some cases multigraph)
Whitney's_planarity_criterion
3-regular graph with no 3-edge-coloring
In the mathematical field of graph theory, a snark is an undirected graph with exactly three edges per vertex whose edges cannot be colored with only three
Snark_(graph_theory)
Combinatorial theory of mechanics and discrete geometry
of the structure. A rigid graph is an embedding of a graph in a Euclidean space which is structurally rigid. That is, a graph is rigid if the structure
Structural_rigidity
Database using graph structures for queries
A graph database (GDB) is a database that uses graph structures for semantic queries with nodes, edges, and properties to represent and store data. A key
Graph_database
Class of artificial neural networks
Graph neural networks (GNNs) are artificial neural networks designed for tasks whose inputs are graphs. Because graphs usually do not have a canonical
Graph_neural_network
Graph made from vertices and edges of a convex polyhedron
such a graph, a representation of it as a subdivision of a convex polygon into smaller convex polygons may be found using the Tutte embedding. Tait conjectured
Polyhedral_graph
Two special graphs in graph theory
mathematical field of graph theory, the Klein graphs are two different but related regular graphs, each with 84 edges. Each can be embedded in the orientable
Klein_graphs
Undirected unit-distance graph requiring four colors
hull of the embedding and every bounded face is a pseudotriangle with only three convex vertices. The complement graph of the Moser graph is a triangle-free
Moser_spindle
Edge-face adjacencies in another graph
with medial graph H are dual to each other. Since the medial graph depends on a particular embedding, the medial graph of a planar graph is not unique;
Medial_graph
Balanced complete multipartite graph
configuration formed by embedding a Turán graph onto the vertices of a regular simplex. An n-vertex graph G is a subgraph of a Turán graph T(n,r) if and only
Turán_graph
On tangency patterns of circles
itself to other objects embedded within the surface. An example comes from dessins d'enfant, a certain type of graph embedding used in algebraic geometry
Circle_packing_theorem
Computer science algorithm
computer science, graph traversal (also known as graph search) refers to the process of visiting (checking and/or updating) each vertex in a graph. Such traversals
Graph_traversal
Undirected graph with 14 vertices
mathematical field of graph theory, the Heawood graph is an undirected graph with 14 vertices and 21 edges, named after Percy John Heawood. The graph is cubic, and
Heawood_graph
a very simple structure as graphs, they are of some importance in topological graph theory because their graph embeddings can still be non-trivial. In
Bouquet_graph
Abstract regular polyhedron with 6 pentagonal faces
of graph theory this is an embedding of the Petersen graph on a real projective plane. With this embedding, the dual graph is K6 (the complete graph with
Hemi-dodecahedron
Type of graph in mathematics and physics
mathematics and physics, a quantum graph is a linear, network-shaped structure of vertices connected on edges (i.e., a graph) in which each edge is given a
Quantum_graph
List of unsolved computational problems
polyhedron in polynomial time? Can a simultaneous embedding with fixed edges for two given graphs be found in polynomial time? Can the square-root sum
List of unsolved problems in computer science
List_of_unsolved_problems_in_computer_science
Finiteness of sets of forbidden graph minors
graph theory, the Robertson–Seymour theorem (also called the graph minors theorem) states that the undirected graphs, partially ordered by the graph minor
Robertson–Seymour_theorem
all other embeddings have greater stretch factor. The graphs that have an embedding with at most a given distortion are closed under graph minor operations
GNRS_conjecture
In geometric graph theory, a convex embedding of a graph is an embedding of the graph into a Euclidean space, with its vertices represented as points and
Convex_embedding
Representation in natural language processing
In natural language processing, a sentence embedding (or document embedding) is a representation of a natural language text as a vector of numbers which
Sentence_embedding
All even-degree subgraphs of a graph
of bounded faces of a planar embedding of its graph. The cycle space of a planar graph is the cut space of its dual graph, and vice versa. The minimum
Cycle_space
Graph describing a topological embedding
In topological graph theory, a graph-encoded map or gem is a method of encoding a cellular embedding of a graph using a different graph with four vertices
Graph-encoded_map
Gluing graphs at complete subgraphs
subset of the other vertices) and vortices (graphs with low pathwidth that replace faces of the surface embedding). These characterizations have been used
Clique-sum
Subgraph of planar graph with Hamiltonian cycle
involving embedding graphs onto universal point sets, simultaneous embedding of multiple graphs, and layered graph drawing. Some classes of planar graphs are
Subhamiltonian_graph
resulting in the expression parallel sheaf of planes. Book embedding, a notion of graph embedding onto sheafs of half-planes Definition on Mathworld Wolfram
Sheaf_of_planes
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