Lipton–Tarjan separator theorem
E583430
The Lipton–Tarjan separator theorem is a fundamental result in graph theory that shows any planar graph can be efficiently divided into roughly equal parts by removing only a relatively small set of vertices, enabling faster algorithms for many computational problems.
All labels observed (4)
| Label | Occurrences |
|---|---|
| A separator theorem for planar graphs | 1 |
| Lipton–Tarjan planar separator theorem | 1 |
| Lipton–Tarjan separator theorem canonical | 1 |
| planar separator theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6316949 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Lipton–Tarjan separator theorem Context triple: [Richard Lipton, knownFor, Lipton–Tarjan separator theorem]
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A.
Tarjan's strongly connected components algorithm
Tarjan's strongly connected components algorithm is a classic linear-time graph algorithm that efficiently identifies all strongly connected components in a directed graph using depth-first search and low-link values.
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B.
Graph Algorithms (book)
"Graph Algorithms" is a foundational textbook by Shimon Even that systematically presents the theory, design, and analysis of algorithms for solving fundamental problems on graphs.
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C.
Furst–Saxe–Sipser lower bounds
Furst–Saxe–Sipser lower bounds are foundational results in circuit complexity theory that established superpolynomial lower bounds for constant-depth Boolean circuits (AC⁰), demonstrating inherent limitations of such circuits for computing certain functions.
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D.
Robert Tarjan
Robert Tarjan is an American computer scientist renowned for his pioneering work in algorithms and data structures, including the development of efficient graph algorithms and the union–find data structure.
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E.
Eppstein
Eppstein is a small historic town in the German state of Hesse, known for its medieval castle and scenic location in the Taunus mountains.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Lipton–Tarjan separator theorem Target entity description: The Lipton–Tarjan separator theorem is a fundamental result in graph theory that shows any planar graph can be efficiently divided into roughly equal parts by removing only a relatively small set of vertices, enabling faster algorithms for many computational problems.
-
A.
Tarjan's strongly connected components algorithm
Tarjan's strongly connected components algorithm is a classic linear-time graph algorithm that efficiently identifies all strongly connected components in a directed graph using depth-first search and low-link values.
-
B.
Graph Algorithms (book)
"Graph Algorithms" is a foundational textbook by Shimon Even that systematically presents the theory, design, and analysis of algorithms for solving fundamental problems on graphs.
-
C.
Furst–Saxe–Sipser lower bounds
Furst–Saxe–Sipser lower bounds are foundational results in circuit complexity theory that established superpolynomial lower bounds for constant-depth Boolean circuits (AC⁰), demonstrating inherent limitations of such circuits for computing certain functions.
-
D.
Robert Tarjan
Robert Tarjan is an American computer scientist renowned for his pioneering work in algorithms and data structures, including the development of efficient graph algorithms and the union–find data structure.
-
E.
Eppstein
Eppstein is a small historic town in the German state of Hesse, known for its medieval castle and scenic location in the Taunus mountains.
- F. None of above. chosen
Statements (44)
| Predicate | Object |
|---|---|
| instanceOf |
separator theorem
ⓘ
theorem in graph theory ⓘ |
| algorithmicAspect | separator can be found in linear time for planar graphs ⓘ |
| appliesTo | planar graphs ⓘ |
| assumption | graph is planar ⓘ |
| balanceProperty | separator splits the graph into roughly equal-sized parts ⓘ |
| citedAs |
Lipton–Tarjan planar separator theorem
ⓘ
linked to:
Lipton–Tarjan separator theorem
|
| complexityImpact |
enables faster algorithms for many NP-hard problems on planar graphs
ⓘ
reduces running time of many planar graph algorithms from O(n^2) to near-linear or n^{3/2} ⓘ |
| componentSizeBound |
at most 2n/3 vertices per component
ⓘ
each component has at most 2n/3 vertices when separator is removed ⓘ |
| enables | divide-and-conquer algorithms on planar graphs ⓘ |
| field | graph theory ⓘ |
| generalizedBy | separator theorems for minor-closed graph classes ⓘ |
| graphClass |
simple planar graphs
ⓘ
undirected graphs ⓘ |
| influenceOn |
graph algorithms textbooks
ⓘ
parameterized complexity on planar graphs ⓘ |
| inspired | subsequent separator theorems in geometric graphs ⓘ |
| mainClaim |
every n-vertex planar graph has a vertex separator of size O(sqrt(n))
ⓘ
the removal of the separator partitions the planar graph into components each with at most a constant fraction of the vertices ⓘ |
| namedAfter |
Richard J. Lipton
ⓘ
linked to:
Richard Lipton
Robert Endre Tarjan ⓘ
linked to:
Robert Tarjan
|
| originalAuthors |
Richard J. Lipton
ⓘ
linked to:
Richard Lipton
Robert Endre Tarjan ⓘ
linked to:
Robert Tarjan
|
| originalTitle |
A separator theorem for planar graphs
ⓘ
linked to:
Lipton–Tarjan separator theorem
|
| proofTechnique |
breadth-first search layering
ⓘ
cycle separators ⓘ planar embedding arguments ⓘ |
| publishedIn | Journal of the ACM ⓘ |
| relatedConcept |
branchwidth
ⓘ
graph separator ⓘ minor-closed graph families ⓘ planar separator theorem ⓘ
linked to:
Lipton–Tarjan separator theorem
treewidth ⓘ |
| separatorSizeBound | O(sqrt(n)) ⓘ |
| separatorType | vertex separator ⓘ |
| usedFor |
design of subquadratic algorithms on planar graphs
ⓘ
planar graph divide-and-conquer dynamic programming ⓘ planar graph maximum independent set approximation ⓘ planar graph recognition algorithms ⓘ planar graph shortest path algorithms ⓘ planar graph vertex cover algorithms ⓘ |
| yearProved | 1979 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Lipton–Tarjan separator theorem Description of subject: The Lipton–Tarjan separator theorem is a fundamental result in graph theory that shows any planar graph can be efficiently divided into roughly equal parts by removing only a relatively small set of vertices, enabling faster algorithms for many computational problems.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Lipton–Tarjan separator theorem
linked to: Lipton–Tarjan separator theorem
linked to: Lipton–Tarjan separator theorem