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See searches and references containing VERTEX SEPARATOR!VERTEX SEPARATOR
Set of graph nodes which separate a given pair of nodes if removed
In graph theory, a vertex subset S ⊂ V {\displaystyle S\subset V} is a vertex separator (or vertex cut, separating set) for nonadjacent vertices a
Vertex_separator
Fundamental unit of which graphs are formed
graph; a vertex separator is a collection of vertices the removal of which would disconnect the remaining graph into small pieces. A k-vertex-connected
Vertex_(graph_theory)
Topics referred to by the same term
as diaphragm Planar separator theorem, a theorem in graph theory Vertex separator, a notion in graph theory Geometric separator, a line that separates
Separator
Any planar graph can be subdivided by removing a few vertices
polynomial expansion. As it is usually stated, the separator theorem states that, in any n {\displaystyle n} -vertex planar graph G = ( V , E ) {\displaystyle
Planar_separator_theorem
Theorem in graph theory
AB-connector of size k is a union of k vertex-disjoint AB-paths. Theorem: The minimum size of an AB-separator is equal to the maximum size of an AB-connector
Menger's_theorem
Graph which remains connected when k or fewer nodes removed
theorem to justify that the minimal-size separator for ( s , t ) {\displaystyle (s,t)} is the number of pairwise vertex-independent paths between them, encode
Vertex_connectivity
Partition of a graph whose components are reachable from all vertices
graphs, a graph is said to be strongly connected if every vertex is reachable from every other vertex. The strongly connected components of a directed graph
Strongly_connected_component
Graph where all long cycles have a chord
) Clearly, this computation depends on chordality. In any graph, a vertex separator is a set of vertices the removal of which leaves the remaining graph
Chordal_graph
Partition of a graph's nodes into 2 disjoint subsets
(graph theory) Graph cuts in computer vision Split (graph theory) Vertex separator Bridge (graph theory) Cutwidth Dicut "NetworkX 2.6.2 documentation"
Cut_(graph_theory)
Method of graph decomposition
of separators, small sets X of vertices in an n-vertex graph such that every X-flap has at most 2n⁄3 vertices. If a graph G does not have a k-vertex separator
Haven_(graph_theory)
Characteristic of undirected graphs
George J., Domke, Gayla S., Miller, Valerie A. (1997), The rank of a graph after vertex addition. Linear Algebra and its Applications, vol. 265, pp. 55–69.
Rank_(graph_theory)
Notation for a polyhedron's vertex figure
variously been called a vertex description, vertex type, vertex symbol, vertex arrangement, vertex pattern, face-vector, vertex sequence. It is also called
Vertex_configuration
Mathematical game
target vertex. But to move one pebble to an adjacent vertex, another pebble at the same vertex must be discarded. Chip-firing game Planar separator theorem
Pebble_game
Type of graph
graph is a connected and "nonseparable" graph, meaning that if any one vertex were to be removed, the graph will remain connected. Therefore a biconnected
Biconnected_graph
Family of graphs whose shallow minors are sparse graphs
is a polynomial. If a hereditary graph family obeys a separator theorem, stating that any n-vertex graph in the family can be split into pieces with at
Bounded_expansion
Partition of a graph by removing fewest possible edges
) 2 {\displaystyle {\frac {n(n-1)}{2}}} minimum cuts. Maximum cut Vertex separator, an analogous concept to minimum cuts for vertices instead of edges
Minimum_cut
Unrelated vertices in graphs
by clique separators", Discrete Mathematics, 55 (2): 221–232, doi:10.1016/0012-365x(85)90051-2. Weisstein, Eric W. "Maximal Independent Vertex Set". MathWorld
Independent set (graph theory)
Independent_set_(graph_theory)
Computational hardness assumption
R. (2008), "Improved approximation algorithms for minimum weight vertex separators", SIAM Journal on Computing, 38 (2): 629–657, doi:10.1137/05064299X
Small set expansion hypothesis
Small_set_expansion_hypothesis
is given by PATH = {⟨D, s, t⟩ | D is a directed graph with a path from vertex s to t}. On a sequential computer, st-connectivity can easily be solved
St-connectivity
Bipartite non-Hamiltonian polyhedral graph
3 , 4 {\displaystyle K_{3,4}} into two symmetric halves by three-vertex separators and then combining one half from each graph. The Herschel graph also
Herschel_graph
Connectivity measure in graph theory
rank zero, while a complete digraph of order n with a self-loop at each vertex has cycle rank n. The cycle rank of a directed graph is closely related
Cycle_rank
the minimum parallel time needed to perform Cholesky decomposition Vertex separator George (1973). Lipton, Rose & Tarjan (1979); Gilbert & Tarjan (1986)
Nested_dissection
Graph representing a permutation
algorithms exploit the fact that the number of inclusion minimal vertex separators in a permutation graph is polynomial in the size of the graph. Permutation
Permutation_graph
Whether one vertex can be reached from another in a graph
with each vertex a relatively small set of so-called separator paths such that any path from a vertex v {\displaystyle v} to any other vertex w {\displaystyle
Reachability
Number denoting a graph's closeness to a tree
vertex is added that is adjacent to all previous vertices, and to take the larger value from the two subgraphs on either side of a clique separator.
Treewidth
Representation of a graph as a path graph "thickened" by some amount
R. (2005), "Improved approximation algorithms for minimum-weight vertex separators", Proc. 37th ACM Symposium on Theory of Computing (STOC 2005), pp
Pathwidth
Polygon that is the boundary of a convex set
common have a separator line. If the polygons are closed and at least one of them is compact, then there are even two parallel separator lines (with a
Convex_polygon
Cycle rank Rank (graph theory) SPQR tree St-connectivity Pixel connectivity Vertex separator Strongly connected component Biconnected graph Bridge v t e
Pixel_connectivity
consists of this vertex and one of its neighbors. A star cutset in a graph G {\displaystyle G} is a vertex separator in which one of the separator vertices is
Skew_partition
Graph whose maximal clique hypergraph is a hypertree
1016/s0195-6698(02)00142-7. De Caria, Pablo; Gutierrez, Marisa (2012), "On Minimal Vertex Separators of Dually Chordal Graphs: Properties and Characterizations", Discrete
Dually_chordal_graph
Topics referred to by the same term
all minimal separators are cliques Dirac's theorem on cycles in k-connected graphs, the result that for every set of k vertices in a k-vertex-connected
Dirac's_theorem
On tangency patterns of circles
intersection graph of a circle packing, called a coin graph, is the graph having a vertex for each circle, and an edge for every pair of circles that are tangent
Circle_packing_theorem
Object in graph theory
from a given root vertex. Given a connected graph G = (V, E) with V the set of vertices and E the set of edges, and with a root vertex r, the level structure
Level_structure
Lowe et al. are those with vertex separators. If the circuit is cut along the qubits corresponding to the vertex separators (with limited number of size)
Quantum_circuit_cutting
graph for n-vertex trees, with only n vertices and O(n log n) edges, and that this is optimal. A construction based on the planar separator theorem can
Universal_graph
Graph minor formed from subgraphs of small diameter
analogously to the planar separator theorem for planar graphs. In particular, if the complete graph Kh is not a d-shallow minor of an n-vertex graph G, then there
Shallow_minor
not change the other separators. As a result, a fixed maximal separator size can be enforced by first calculating all separator sizes and then iteratively
Decomposition method (constraint satisfaction)
Decomposition_method_(constraint_satisfaction)
Type of directed graph
metric space, such as the Euclidean distance in the plane. The NNG has a vertex for each point, and a directed edge from p to q whenever q is a nearest
Nearest_neighbor_graph
Polygon in which all angles are right
all of whose sides meet at right angles. Thus the interior angle at each vertex is either 90° or 270°. Rectilinear polygons are a special case of isothetic
Rectilinear_polygon
In applied mathematics, a technique to find the shortest path
at, at query time. To achieve this, iterative vertex contractions are performed. When contracting a vertex v {\displaystyle v} it is temporarily removed
Contraction_hierarchies
Tree graph with all nodes within distance 1 from central path
there exists a path that contains every vertex of degree two or more. They are the trees in which every vertex of degree at least three has at most two
Caterpillar_tree
, v 1 , v 2 , … {\displaystyle v_{0},v_{1},v_{2},\dots } in which each vertex appears at most once in the sequence and each two consecutive vertices in
End_(graph_theory)
Task of computing complete subgraphs
single vertex or even the empty set), grow the current clique one vertex at a time by looping through the graph's remaining vertices. For each vertex v that
Clique_problem
(n^{2})} 3-separators and Θ ( n 4 ) {\displaystyle \Theta (n^{4})} non-laminar pairs of 3-separators, making it inefficient to enumerate all separators and then
Laminar_set_family
Graph that can be embedded in the plane
intersection graph of line segments in the plane. The planar separator theorem states that every n-vertex planar graph can be partitioned into two subgraphs of
Planar_graph
Graph which can be made planar by removing a single node
is a graph that can be made planar by the removal of a single vertex. The deleted vertex is called an apex of the graph. It is an apex, not the apex because
Apex_graph
Number of planar subgraphs to cover a graph
S2CID 31670574. Christian A. Duncan, On Graph Thickness, Geometric Thickness, and Separator Theorems, CCCG 2009, Vancouver, BC, August 17–19, 2009 Ringel, Gerhard
Thickness_(graph_theory)
Gluing graphs at complete subgraphs
other contexts, such as the SPQR-tree decomposition of graphs into their 3-vertex-connected components, all edges should be removed. And in yet other contexts
Clique-sum
Geometric inequality applicable to any closed curve
Isoperimetric point List of triangle inequalities Mixed volume Planar separator theorem Blåsjö, Viktor (2005). "The Evolution of the Isoperimetric Problem"
Isoperimetric_inequality
Subgraph with contracted edges
H-minor-free graphs have a separator theorem similar to the planar separator theorem for planar graphs: for any fixed H, and any n-vertex H-minor-free graph G
Graph_minor
Graphs whose distances obey Ptolemy's inequality
overlapping maximal cliques, the intersection of the two cliques is a separator that splits the differences of the two cliques. In the illustration of
Ptolemaic_graph
Algorithmic problem of finding non-crossing drawings
permutations of cyclic edge-order for planar embeddings of biconnected components. Vertex addition methods work by maintaining a data structure representing the possible
Planarity_testing
Size of largest complete graph made by contracting edges of a given graph
vertex is added that is adjacent to all previous vertices, and to take the larger value from the two subgraphs on either side of a clique separator.
Hadwiger_number
Complexity class
version) Subgraph isomorphism problem Subset sum problem Clique problem Vertex cover problem Independent set problem Dominating set problem Graph coloring
NP-completeness
Abstraction of linear independence of vectors
is a separator that is neither E nor the empty set. An irreducible separator is a non-empty separator that contains no other non-empty separator. The
Matroid
Graph formed by subdivision of triangles
on four vertices, formed by choosing any vertex and its three earlier neighbors. Every minimal clique separator (a clique that partitions the graph into
Apollonian_network
Intersection graph for curves in the plane
there exists a set of curves, or strings, such that the graph having a vertex for each curve and an edge for each intersecting pair of curves is isomorphic
String_graph
Graph theory model
by starting with a (k + 1)-vertex complete graph and then repeatedly adding vertices in such a way that each added vertex v has exactly k neighbors U
K-tree
Graph with at most one crossing per edge
have the same color, no two adjacent faces have the same color, and no vertex and face that are adjacent to each other have the same color. This can obviously
1-planar_graph
British mathematician
significant results: (with Noga Alon) a separator theorem for graphs with an excluded minor, extending the planar separator theorem of Richard Lipton and Robert
Paul_Seymour_(mathematician)
Point in a triangle that can be seen as its middle under some criteria
the coordinates refer to the (c, b, a) triangle and (using "|" as the separator) the reflection of an arbitrary point γ : β : α {\displaystyle \gamma
Triangle_center
Point set triangulation minimizing total length
point set must be subdivided into triangles that meet edge-to-edge and vertex-to-vertex, in such a way as to minimize the sum of the perimeters of the triangles
Minimum-weight_triangulation
Standard UNIX utility
interchangeability of white space separators so the following inputs are equivalent: Pairs of identical items indicate presence of a vertex, but not ordering (so
Tsort
Invariant in graph theory
same queue, then it should not be possible to have a < c < d < b in the vertex ordering. The queue number qn(G) of a graph G is the minimum number of queues
Queue_number
Subdivision of vertices into disjoint sets
finite approximation factor unless P = NP. The planar separator theorem states that any n-vertex planar graph can be partitioned into roughly equal parts
Graph_partition
reparative, separability, separable, separate, separation, separative, separator, separatory, separatrix, sever, severability, severable, several, severance
List of Greek and Latin roots in English/P–Z
List_of_Greek_and_Latin_roots_in_English/P–Z
the SI writing style, a non-breaking space can be used as a thousands separator, i.e., to separate the digits of a number at every power of 1000. Multiples
1000_(number)
Chordal graph with the given graph as a subgraph
completion of a given 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
Chordal_completion
Graph layout on multiple half-planes
exact book thickness of a given graph, with or without knowing a fixed vertex ordering along the spine of the book. Testing the existence of a three-page
Book_embedding
Tiling of the hyperbolic plane
Jana; van Leeuwen, Erik Jan; Walczak, Bartosz; Wegrzycki, Karol (2024). "Separator theorem and algorithms for planar hyperbolic graphs". In Mulzer, Wolfgang;
Binary_tiling
topological graph cross a finite number of times, no edge passes through a vertex different from its endpoints, and no two edges touch each other (without
Topological_graph
Family of connectors typically used for analog signals
Headsets using this wiring are sometimes indicated by black plastic separators between the rings. The CTIA/AHJ standard reverses these contacts, putting
Phone_connector_(audio)
Graph drawing with vertices on a circle
interaction graph, or natural subgroups within a social network. If multiple vertex circles are used in this way, other methods such as force-directed graph
Circular_layout
Network that allows computers to share resources and communicate with each other
theoretical results (e.g., [16]) which show that, by optimally placing separators, i.e., elements that connect levels in the hierarchy, tremendous gain
Computer_network
Units that are not part of a coherent system
the unit newton), that is in the number of nines following the decimal separator in writing the number in question. For example, "three nines" or "3N"
List of non-coherent units of measurement
List_of_non-coherent_units_of_measurement
List of experiments at CERN, Switzerland
Synchrotron SPS: Super Proton Synchrotron ISOLDE: On-Line Isotope Mass Separator ISR: Intersecting Storage Rings LEP: Large Electron–Positron Collider
List of Super Proton Synchrotron experiments
List_of_Super_Proton_Synchrotron_experiments
Perfect graph theorem (graph theory) Perlis theorem (graph theory) Planar separator theorem (graph theory) Pólya enumeration theorem (combinatorics) Ramsey's
List_of_theorems
Logical formulation of graph properties
sentence can be interpreted as meaning that for every vertex u {\displaystyle u} there is another vertex v {\displaystyle v} that is adjacent to u {\displaystyle
Logic_of_graphs
Polygon through a set of points
polygonalizations with the additional constraint that they make a right turn at every vertex, if they exist, are uniquely determined. Each axis-parallel line through
Polygonalization
Partition of Earth's surface into subdivided cells
with similar precision, but differ in string-length, separators-use and alphabet (non-separator characters). In some cases the "original DGG" representation
Discrete_global_grid
Set of primitive shapes whose union equals a polygon
continuators or separators; thus, such a polygon is analogous to a tree graph. A general polygon is analogous to a general graph. Just like the vertex cover problem
Polygon_covering
Hutchinson (born 1945), American graph theorist who extended the planar separator theorem to graphs of higher genus Marie Hušková (born 1942), Czech mathematician
List_of_women_in_mathematics
Undirected graph with 11 nodes and 27 edges
maximal planar graph. As with every maximal planar graph, it is also 3-vertex-connected: the removal of any two of its vertices leaves a connected subgraph
Goldner–Harary_graph
1995 edition of the Fortran programming language standard
record 10 6.4 (1.0,0.0) (2.0,0.0) t test/ (in which blanks are used as separators), then i, a, field, flag, and title will acquire the values 10, 6.4, (1
Fortran_95_language_features
reparable, reparation, reparative, separable, separate, separation, separator, sever, severable, several, severance pascō pasc- pav- past- feed antepast
List of Latin verbs with English derivatives
List_of_Latin_verbs_with_English_derivatives
Class of chemical compounds
"Preparation of Ion-molecules of the Inert Gases in an Electromagnetic Isotope Separator". Nature. 201 (4914): 69–70. Bibcode:1964Natur.201...69F. doi:10.1038/201069a0
Argon_compounds
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