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Any planar graph can be subdivided by removing a few vertices
In graph theory, the planar separator theorem is a form of isoperimetric inequality for planar graphs, that states that any planar graph can be split into
Planar_separator_theorem
Graph that can be embedded in the plane
a theorem) states that every planar graph can be represented as an intersection graph of line segments in the plane. The planar separator theorem states
Planar_graph
On tangency patterns of circles
applications in conformal mapping, the construction of polyhedra, planar separator theorems, graph drawing, and the theory of random walks. The study of the
Circle_packing_theorem
Topics referred to by the same term
known as diaphragm Planar separator theorem, a theorem in graph theory Vertex separator, a notion in graph theory Geometric separator, a line that separates
Separator
Index of articles associated with the same name
logically equivalent "past → future" form. Planar separator theorem (graph theory) states that any planar graph can be split into smaller pieces by removing
Separation_theorem
Set of graph nodes which separate a given pair of nodes if removed
for applications in computer science, such as the planar separator theorem. Let S be an (a,b)-separator, that is, a vertex subset that separates two nonadjacent
Vertex_separator
Complexity class
dominating set problems for planar graphs are NP-complete, but can be solved in subexponential time using the planar separator theorem. "Each instance of an
NP-completeness
Mathematical game
Planar Separator Theorem, SIAM J. Comput. 1980 Noga Alon, [[Paul Seymour (mathematician)|]], [[Robin Thomas (mathematician)|]], A Separator Theorem for
Pebble_game
Topics referred to by the same term
Personal Storage Table, a file format used in Microsoft applications Planar separator theorem, in graph theory Pocket set theory, in mathematics Post-stall technology
PST
resulting matrix has O(n log n) nonzeros, due to the planar separator theorem guaranteeing separators of size O(√n). For arbitrary graphs there is a nested
Nested_dissection
Graph minor formed from subgraphs of small diameter
excluded shallow minors can be partitioned analogously to the planar separator theorem for planar graphs. In particular, if the complete graph Kh is not a
Shallow_minor
the planar separator theorem can be used to show that n-vertex planar graphs have universal graphs with O(n3/2) edges, and that bounded-degree planar graphs
Universal_graph
Ore's theorem (graph theory) Paley's theorem (algebra) Perfect graph theorem (graph theory) Perlis theorem (graph theory) Planar separator theorem (graph
List_of_theorems
American computer scientist (born 1946)
Prize winner, 2014 SL (complexity) Take-grant protection model Planar separator theorem Richard Lipton at the Mathematics Genealogy Project Lipton, R (1975)
Richard_Lipton
in geometric graphs. The planar separator theorem may be proven by using the circle packing theorem to represent a planar graph as the contact graph
Geometric_separator
Whether one vertex can be reached from another in a graph
proof that such separators can always be found is related to the Planar Separator Theorem of Lipton and Tarjan, and these separators can be located in
Reachability
Intersection graph for curves in the plane
forbidden induced minors for string graphs. Analogously to the planar separator theorem, every m {\displaystyle m} -edge string graph can be partitioned
String_graph
Mathematical propositions in network flow theory
In graph theory, approximate max-flow min-cut theorems concern the relationship between the maximum flow rate (max-flow) and the minimum cut (min-cut)
Approximate max-flow min-cut theorem
Approximate_max-flow_min-cut_theorem
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
Additionally, the 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
Graph_minor
Indian computer scientist and mathematician
such as Minkowski's theorem, sphere packing, the representation of planar graphs by tangent circles, the planar separator theorem. The second section
Pankaj_K._Agarwal
Graph layout on multiple half-planes
Because graphs of book thickness two are planar graphs, they obey the planar separator theorem: they have separators, subsets of vertices whose removal splits
Book_embedding
Graph with at most one crossing per edge
parameters are bounded. In contrast to Fáry's theorem for planar graphs, not every 1-planar graph may be drawn 1-planarly with straight line segments for its edges
1-planar_graph
Task of computing complete subgraphs
S2CID 6390600. Lipton, R. J.; Tarjan, R. E. (1980), "Applications of a planar separator theorem", SIAM Journal on Computing, 9 (3): 615–627, doi:10.1137/0209046
Clique_problem
Algorithmic problem of finding non-crossing drawings
subgraph if it is not. Planarity testing algorithms typically take advantage of theorems in graph theory that characterize the set of planar graphs in terms
Planarity_testing
British mathematician
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)
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
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
Gluing graphs at complete subgraphs
and maximal planar graphs without deleting edges. The graphs in which every induced cycle of length four or greater forms a minimal separator of the graph
Clique-sum
Polygon through a set of points
{\displaystyle 54.6^{n}} polygonalizations. Methods applying the planar separator theorem to labeled triangulations of the points can be used to count all
Polygonalization
laminar set family. Laminar families of separators play a crucial role in decomposing planar graphs. A k-separator (or k-cutset) in a k-connected graph is
Laminar_set_family
Chordal graph with the given graph as a subgraph
planar graphs may arise in the solution of two-dimensional finite element systems; it follows from the planar separator theorem that every planar graph
Chordal_completion
Concept in computational geometry
rectangles. Ravi, S. S.; Hunt, H. B. (1987). "An application of the planar separator theorem to counting problems". Information Processing Letters. 25 (5):
Maximum_disjoint_set
Representation of a graph as a path graph "thickened" by some amount
which a similar separator theorem holds. Since, like planar graphs, the graphs in any fixed minor-closed graph family have separators of size O(√n), it
Pathwidth
Number of planar subgraphs to cover a graph
Christian A. Duncan, On Graph Thickness, Geometric Thickness, and Separator Theorems, CCCG 2009, Vancouver, BC, August 17–19, 2009 Ringel, Gerhard (1959)
Thickness_(graph_theory)
American mathematician
generalizing the planar separator theorem to surfaces.[GHT84] With S. Wagon she has co-authored papers on algorithmic aspects of the four color theorem.[HW98] Albertson
Joan_Hutchinson
Family of graphs whose shallow minors are sparse graphs
sublinear separator theorems. Because of the connection between separators and expansion, every minor-closed graph family, including the family of planar graphs
Bounded_expansion
Number denoting a graph's closeness to a tree
take the larger value from the two subgraphs on either side of a clique separator. The set of all such functions forms a complete lattice under the operations
Treewidth
Graph which remains connected when k or fewer nodes removed
of nonadjacent nodes to disconnect, using Menger's theorem to justify that the minimal-size separator for ( s , t ) {\displaystyle (s,t)} is the number
Vertex_connectivity
Object in graph theory
MR 0440883. Lipton, Richard J.; Tarjan, Robert E. (1979), "A separator theorem for planar graphs", SIAM Journal on Applied Mathematics, 36 (2): 177–189
Level_structure
Graph where all long cycles have a chord
graph, a vertex separator is a set of vertices the removal of which leaves the remaining graph disconnected. According to a theorem of Dirac (1961),
Chordal_graph
Graph formed by subdivision of triangles
clique separators have the same size is a k-tree, and Apollonian networks are examples of 3-trees. Not every 3-tree is planar, but the planar 3-trees
Apollonian_network
Partition of a graph by removing fewest possible edges
{\displaystyle {\frac {n(n-1)}{2}}} minimum cuts. Maximum cut Vertex separator, an analogous concept to minimum cuts for vertices instead of edges "4
Minimum_cut
Abstraction of linear independence of vectors
Kuratowski's theorem, the dual of a graphic matroid M {\displaystyle M} is a graphic matroid if and only if M {\displaystyle M} is the matroid of a planar graph
Matroid
k-quasi-planar if it has no k pairwise crossing edges. Using this terminology, if a topological graph is 2-quasi-planar, then it is a planar graph. It
Topological_graph
Size of largest complete graph made by contracting edges of a given graph
series–parallel graph. Wagner's theorem, which characterizes the planar graphs by their forbidden minors, implies that the planar graphs have Hadwiger number
Hadwiger_number
Type of directed graph
special case of the k-NNG, namely it is the 1-NNG. k-NNGs obey a separator theorem: they can be partitioned into two subgraphs of at most n(d + 1)/(d
Nearest_neighbor_graph
Undirected graph with 11 nodes and 27 edges
Both of its maximal cliques and clique separators have the same size, hence the graph is chordal. As a planar 3-tree, it forms an example of an Apollonian
Goldner–Harary_graph
Graph which can be made planar by removing a single node
theory, a branch of mathematics, an apex graph is a graph that can be made planar by the removal of a single vertex. The deleted vertex is called an apex
Apex_graph
Tiling of the hyperbolic plane
Erik Jan; Walczak, Bartosz; Wegrzycki, Karol (2024). "Separator theorem and algorithms for planar hyperbolic graphs". In Mulzer, Wolfgang; Phillips, Jeff
Binary_tiling
Unrelated vertices in graphs
example for that. Another important tool are clique separators as described by Tarjan. Kőnig's theorem implies that in a bipartite graph the maximum independent
Independent set (graph theory)
Independent_set_(graph_theory)
Method of graph decomposition
graph. Even faster algorithms are possible for graphs with few minimal separators. Bodlaender, Grigoriev, and Koster studied heuristics for finding brambles
Bramble_(graph_theory)
Bipartite non-Hamiltonian polyhedral graph
exchanges pairs of degree-three vertices. By Steinitz's theorem, every graph that is planar and 3-vertex-connected is the skeleton of some convex polyhedron
Herschel_graph
Award for advancements in discrete mathematics
Appel and Wolfgang Haken for the four color theorem. Paul Seymour for generalizing the max-flow min-cut theorem to matroids. 1982: D.B. Judin, Arkadi Nemirovski
Fulkerson_Prize
American computer scientist and educator
problems. The paper was part of a line of work connecting network flows, separators, and approximation guarantees for difficult graph problems. Rao has also
Satish_B._Rao
Invariant in graph theory
b/2\rceil \}} respectively. Every 1-queue graph is a planar graph, with an "arched leveled" planar embedding in which the vertices are placed on parallel
Queue_number
Notation for a polyhedron's vertex figure
pq. This notation applies to polygonal tilings as well as polyhedra. A planar vertex configuration denotes a uniform tiling just like a nonplanar vertex
Vertex_configuration
Chemical element with atomic number 118 (Og)
reaches the next chamber, the separator; if a new nucleus is produced, it is carried with this beam. In the separator, the newly produced nucleus is
Oganesson
Passive electronic component providing electrical resistance
scheme is the RKM code following IEC 60062. Rather than using a decimal separator, this notation uses a letter loosely associated with SI prefixes corresponding
Resistor
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
Calculation commonly applied to engines and mobile power sources
the dielectric medium to nanopores and a very thin high permittivity separator. While capacitors tend not to be as temperature-sensitive as batteries
Power-to-weight_ratio
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