Menger theorem in graph theory
E735111
Menger's theorem in graph theory is a fundamental result that characterizes the connectivity between two vertices in a graph by equating the maximum number of pairwise internally disjoint paths between them with the minimum size of a vertex cut separating them.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Menger theorem in graph theory canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T8454639 — 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: Menger theorem in graph theory Context triple: [Karl Menger, notableFor, Menger theorem in graph theory]
-
A.
Erdős–Gallai theorem
The Erdős–Gallai theorem is a fundamental result in graph theory that characterizes which sequences of nonnegative integers can occur as the degree sequences of simple graphs.
-
B.
Lipton–Tarjan separator theorem
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.
-
C.
Erdős on Graphs: His Legacy
Erdős on Graphs: His Legacy is a mathematical monograph by Fan Chung and Ronald Graham that surveys and extends Paul Erdős’s influential work in graph theory and combinatorics.
-
D.
Erdős–Ko–Rado theorem
The Erdős–Ko–Rado theorem is a fundamental result in extremal combinatorics that determines the maximum size of a family of subsets of a finite set in which every pair of subsets has a non-empty intersection.
-
E.
Erdős–Stone theorem
The Erdős–Stone theorem is a fundamental result in extremal graph theory that asymptotically determines the maximum number of edges in an n-vertex graph that avoids containing a given subgraph.
- 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: Menger theorem in graph theory Target entity description: Menger's theorem in graph theory is a fundamental result that characterizes the connectivity between two vertices in a graph by equating the maximum number of pairwise internally disjoint paths between them with the minimum size of a vertex cut separating them.
-
A.
Erdős–Gallai theorem
The Erdős–Gallai theorem is a fundamental result in graph theory that characterizes which sequences of nonnegative integers can occur as the degree sequences of simple graphs.
-
B.
Lipton–Tarjan separator theorem
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.
-
C.
Erdős on Graphs: His Legacy
Erdős on Graphs: His Legacy is a mathematical monograph by Fan Chung and Ronald Graham that surveys and extends Paul Erdős’s influential work in graph theory and combinatorics.
-
D.
Erdős–Ko–Rado theorem
The Erdős–Ko–Rado theorem is a fundamental result in extremal combinatorics that determines the maximum size of a family of subsets of a finite set in which every pair of subsets has a non-empty intersection.
-
E.
Erdős–Stone theorem
The Erdős–Stone theorem is a fundamental result in extremal graph theory that asymptotically determines the maximum number of edges in an n-vertex graph that avoids containing a given subgraph.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
result in combinatorics
ⓘ
theorem in graph theory ⓘ |
| appliesTo |
finite graphs
ⓘ
undirected graphs ⓘ |
| assumes | simple undirected graph in standard formulations ⓘ |
| characterizes | connectivity between two vertices in a graph ⓘ |
| equates |
maximum number of pairwise internally vertex-disjoint paths between two vertices
ⓘ
minimum size of a vertex separator between the two vertices ⓘ |
| field |
combinatorics
ⓘ
graph theory ⓘ |
| hasEdgeVersionStatement | for any two distinct vertices s and t, the maximum number of pairwise edge-disjoint s–t paths equals the minimum size of an s–t edge cut ⓘ |
| hasFormulation |
edge version
ⓘ
global vertex version ⓘ local vertex version ⓘ |
| hasGeneralization | infinite graphs (with additional conditions) ⓘ |
| hasGlobalVertexVersionStatement | the vertex connectivity of a finite graph equals the minimum, over all pairs of distinct vertices, of the size of a smallest vertex cut separating them ⓘ |
| hasLocalVertexVersionStatement | for any two nonadjacent vertices s and t, the maximum number of pairwise internally vertex-disjoint s–t paths equals the minimum size of an s–t vertex cut ⓘ |
| hasProofMethod |
combinatorial arguments
ⓘ
reduction to max-flow min-cut (modern proofs) ⓘ |
| implies |
a graph is k-vertex-connected if and only if every pair of vertices has at least k internally vertex-disjoint paths between them
ⓘ
edge connectivity equals minimum size of an edge cut separating some pair of vertices ⓘ vertex connectivity equals minimum size of a vertex cut separating some pair of vertices ⓘ |
| influenced |
development of connectivity theory in graphs
ⓘ
network flow theory ⓘ |
| involvesConcept |
internally disjoint paths
ⓘ
separating set of edges ⓘ separating set of vertices ⓘ |
| isTaughtIn |
advanced undergraduate graph theory courses
ⓘ
graduate combinatorics courses ⓘ |
| namedAfter | Karl Menger NERFINISHED ⓘ |
| publishedIn | Fundamenta Mathematicae NERFINISHED ⓘ |
| relatedTo |
Whitney's theorem on 2-connectivity
NERFINISHED
ⓘ
edge connectivity ⓘ max-flow min-cut theorem NERFINISHED ⓘ vertex connectivity ⓘ |
| relates |
maximum number of pairwise internally vertex-disjoint paths
ⓘ
minimum size of a vertex cut separating two vertices ⓘ |
| topic |
disjoint paths
ⓘ
edge connectivity ⓘ edge cuts ⓘ graph connectivity ⓘ vertex connectivity ⓘ vertex cuts ⓘ |
| usedIn |
flow networks
ⓘ
graph algorithms ⓘ network reliability theory ⓘ routing and communication networks ⓘ |
| yearProved | 1927 ⓘ |
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: Menger theorem in graph theory Description of subject: Menger's theorem in graph theory is a fundamental result that characterizes the connectivity between two vertices in a graph by equating the maximum number of pairwise internally disjoint paths between them with the minimum size of a vertex cut separating them.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Karl Menger