Width of a hypergraph¶

In graph theory, there are two related properties of a hypergraph that are called its "width".

Version: v1 · 2026-09-28 · History
Domain-specific #: 12884
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Hypergraph Theory, Graph Theory → Mathematics

Core Idea¶

Width of a hypergraph is treated here as the recurring mathematics_logic_statistics identity summarized by this source-grounded definition: In graph theory, there are two related properties of a hypergraph that are called its "width".

of the graph can be pinned by a single edge. Here, a matching is shown in red, and an edge that pins it in yellow. In graph theory, there are two related properties of a hypergraph that are called its "width".

Given a hypergraph H = (V, E), we say that a set K of edges pins another set F of edges if every edge in F intersects some edge in K. The width of H, denoted w(H), is the smallest size of a subset of E that pins E. The matching width of H, denoted mw(H), is the maximum, over all matchings M in H, of the minimum size of a subset of E that pins M.

For Width of a hypergraph, the abstraction is narrower than the article's general subject matter: a positive case must preserve In graph theory, there are two related properties of a hypergraph that are called its "width". Retaining only the name, a familiar example, or a downstream effect is insufficient. The specialist roles and tests remain anchored in mathematics_logic_statistics, which is why this identity is domain-specific rather than prime.

Structural Signature¶

Sig role-phrases:

Defining carrier — w(H) = 2, since E is pinned e.g. by the set { {A,a}, {B,b} }, and cannot be pinned by any smaller set.
Constitutive relation — mw(H) = 1, since every matching can be pinned by a single edge.
Operating condition — There are two matchings: is pinned e.g. by { {A,b} }, and { {A,b}, {B,a} } is pinned e.g. by { {A, a} }.
Recognition evidence — This is because w(H) > r iff H has no pinning-set of size r, iff for every subset of r edges of H there is an edge that is not pinned by it, iff every subset of r edges of H has a common neighbor in D(H).
Admissible variation — of the graph can be pinned by a single edge.
Characteristic consequence — Let H be the hypergraph with vertex set V = {A,B; a,b} and edge set: E = { {A,a}, {B,b}, {A,b}, {B,a} } The widths of H are.
Failure boundary — The disjointness graph of H, denoted D(H), is a graph where each edge in H is a vertex in D(H), and every two disjoint edges in H are adjacent in D(H).

What It Is Not¶

Not the whole field of mathematics_logic_statistics. The node requires the specific identity stated by In graph theory, there are two related properties of a hypergraph that are called its "width".
Not an over-broad reading. w(H) > r if and only if for every set of r vertices in L(H) there is a vertex not adjacent to any of them.
Not an over-broad reading. This is because w(H) > r iff H has no pinning-set of size r, iff for every subset of r edges of H there is an edge that is not pinned by it, iff every subset of r edges of H has a common neighbor in D(H).
Not an over-broad reading. mw(H) > r if and only if for every set of r vertices in L(H) there is a vertex not adjacent to any of them, and in addition, there is an independent set I in L(H) which contains a vertex not adjacent to any such set.
Not automatically Line graph of a hypergraph. Retrieval proximity does not establish equivalence; the two identities must be compared by carrier, operation, and failure boundary.

Scope of Application¶

Width of a hypergraph applies literally inside mathematics_logic_statistics wherever the source-defined carrier and relation can be established. Its documented habitats include:

Documented setting. The width of a hypergraph is used in Hall-type theorems for hypergraphs.
Examples. Let H be the hypergraph with vertex set V = {A,B; a,b} and edge set: E = { {A,a}, {B,b}, {A,b}, {B,a} } The widths of H are.
Examples. w(H) = 2, since E is pinned e.g. by the set { {A,a}, {B,b} }, and cannot be pinned by any smaller set.
Examples. mw(H) = 1, since every matching can be pinned by a single edge.
Examples. There are two matchings: is pinned e.g. by { {A,b} }, and { {A,b}, {B,a} } is pinned e.g. by { {A, a} }.
Characterizations. The disjointness graph of H, denoted D(H), is a graph where each edge in H is a vertex in D(H), and every two disjoint edges in H are adjacent in D(H).

Outside mathematics_logic_statistics, the name should be retained only when these same operational conditions survive; otherwise the comparison belongs to the broader parent Classification or should be marked as analogy.

Clarity¶

A clear use of Width of a hypergraph names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In graph theory, there are two related properties of a hypergraph that are called its "width". The strongest recognition evidence in the frozen account is: This is because w(H) > r iff H has no pinning-set of size r, iff for every subset of r edges of H there is an edge that is not pinned by it, iff every subset of r edges of H has a common neighbor in D(H). A report should distinguish that evidence from a proxy, consequence, or common implementation. It should also state the qualification w(H) > r if and only if for every set of r vertices in L(H) there is a vertex not adjacent to any of them. so that a reader can reproduce the classification rather than infer it from topical resemblance.

Manages Complexity¶

Width of a hypergraph compresses multiple mathematics_logic_statistics details into a stable diagnostic relation. The source shows both the central mechanism—mw(H) = 1, since every matching can be pinned by a single edge.—and the practical consequence—let H be the hypergraph with vertex set V = {A,B; a,b} and edge set: E = { {A,a}, {B,b}, {A,b}, {B,a} } The widths of H are. This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit. It is lossy by design: local history and implementation details may be omitted only when they do not alter the defining relation.

Abstract Reasoning¶

Type the carrier. Identify the mathematics_logic_statistics entities to which the claim applies.
State the relation. Use the source-grounded identity: In graph theory, there are two related properties of a hypergraph that are called its "width".
Check operation and conditions. There are two matchings: is pinned e.g. by { {A,b} }, and { {A,b}, {B,a} } is pinned e.g. by { {A, a} }.
Demand recognition evidence. This is because w(H) > r iff H has no pinning-set of size r, iff for every subset of r edges of H there is an edge that is not pinned by it, iff every subset of r edges of H has a common neighbor in D(H).
Test variation. Change an implementation or setting while preserving of the graph can be pinned by a single edge.
Run the collapse test. Remove the defining operation; if the label still seems equally apt, only a topic or correlate was retained.
Reduce cautiously. When the specialist conditions cannot be carried, route the residual comparison to Classification.

Knowledge Transfer¶

Within the home domain. Knowledge about Width of a hypergraph transfers literally when a new case preserves the same carrier type, relation, and recognition test. The width of a hypergraph is used in Hall-type theorems for hypergraphs. Let H be the hypergraph with vertex set V = {A,B; a,b} and edge set: E = { {A,a}, {B,b}, {A,b}, {B,a} } The widths of H are.

Beyond the home domain. No canonical parent is asserted for Width of a hypergraph. An outside case receives the specialist name only when the same typed roles and rejection conditions can be filled literally; otherwise the comparison remains an analogy pending later graph densification.

Examples¶

Canonical¶

w(H) = 2, since E is pinned e.g. by the set { {A,a}, {B,b} }, and cannot be pinned by any smaller set. This case is canonical because it supplies a concrete carrier and lets the defining relation be checked rather than merely named.

Mapped back: carrier → the entities in the documented case; operation → In graph theory, there are two related properties of a hypergraph that are called its "width"; recognition evidence → This is because w(H) > r iff H has no pinning-set of size r, iff for every subset of r edges of H there is an edge that is not pinned by it, iff every subset of r edges of H has a common neighbor in D(H)

Applied / In Practice¶

There are two matchings: is pinned e.g. by { {A,b} }, and { {A,b}, {B,a} } is pinned e.g. by { {A, a} }. The applied case shows how the identity is used under a second setting or qualification while keeping the same operative relation.

Mapped back: changed setting → Examples; invariant → In graph theory, there are two related properties of a hypergraph that are called its "width"; boundary → the case exits the class when w(H) > r if and only if for every set of r vertices in L(H) there is a vertex not adjacent to any of them

Structural Tensions¶

T1 — Stable identity versus admissible variation. w(H) > r if and only if for every set of r vertices in L(H) there is a vertex not adjacent to any of them. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Which changes preserve the defining relation, and which replace it?

T2 — Recognition versus proxy. This is because w(H) > r iff H has no pinning-set of size r, iff for every subset of r edges of H there is an edge that is not pinned by it, iff every subset of r edges of H has a common neighbor in D(H). The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the cited evidence establish the identity or only a correlated sign?

T3 — Definition versus implementation. mw(H) > r if and only if for every set of r vertices in L(H) there is a vertex not adjacent to any of them, and in addition, there is an independent set I in L(H) which contains a vertex not adjacent to any such set. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Is the observed implementation constitutive, optional, or merely common?

T4 — Scope versus overextension. Let H be the hypergraph with vertex set V = {A,B; a,b} and edge set: E = { {A,a}, {B,b}, {A,b}, {B,a} } The widths of H are. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Can every claimed application fill the same typed roles without metaphor?

T5 — Transfer versus domain accent. w(H) = 2, since E is pinned e.g. by the set { {A,a}, {B,b} }, and cannot be pinned by any smaller set. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the receiving case instantiate Width of a hypergraph literally, co-instantiate Classification, or only resemble it?

T6 — Autonomy versus reduction. mw(H) = 1, since every matching can be pinned by a single edge. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: What does Width of a hypergraph distinguish that the broader parent Classification leaves together?

Structural–Framed Character¶

Width of a hypergraph is structural-leaning. Its structural side is the repeatable organization summarized by In graph theory, there are two related properties of a hypergraph that are called its "width". Its framed side is the mathematics_logic_statistics vocabulary that fixes the carrier, evidence, exceptions, and admissible transformations.

Evaluative weight: the identity can be stated descriptively even when applications carry practical stakes. Human-practice dependence: the source-grounded carrier determines whether the relation exists independently or is constituted by a practice. Institutional origin: disciplinary conventions stabilize the name and test. Vocabulary portability: There are two matchings: is pinned e.g. by { {A,b} }, and { {A,b}, {B,a} } is pinned e.g. by { {A, a} }. Import versus recognition: literal transfer requires the same mechanism; shape alone is analogy.

Its portable skeleton is Classification. Its character: a recurring specialist identity whose thin organization can be abstracted, while its operational meaning remains domain-bound.

Structural Core vs. Domain Accent¶

What is skeletal. In graph theory, there are two related properties of a hypergraph that are called its "width". The stable skeleton is the typed relation expressed in that definition and the entry's recognition and collapse tests. The source identifies these operative conditions: w(H) = 2, since E is pinned e.g. by the set { {A,a}, {B,b} }, and cannot be pinned by any smaller set. mw(H) = 1, since every matching can be pinned by a single edge. It further constrains recognition and variation through: There are two matchings: is pinned e.g. by { {A,b} }, and { {A,b}, {B,a} } is pinned e.g. by { {A, a} }. This is because w(H) > r iff H has no pinning-set of size r, iff for every subset of r edges of H there is an edge that is not pinned by it, iff every subset of r edges of H has a common neighbor in D(H).

What is domain-bound. mathematics logic statistics supplies the operative entities, technical vocabulary, warrants, and exceptions that make Width of a hypergraph literal. Its documented scope includes the condition that The width of a hypergraph is used in Hall-type theorems for hypergraphs. Another bounded application condition is that Let H be the hypergraph with vertex set V = {A,B; a,b} and edge set: E = { {A,a}, {B,b}, {A,b}, {B,a} } The widths of H are. These are not decorative examples; they determine which carrier and evidence can fill the abstraction's roles.

Why no parent is asserted. Removing those specialist details does not currently yield one live catalog node that is a necessary genus for every instance. The entry is therefore approved as unparented rather than attached by topical resemblance. Its collapse evidence remains specific—of the graph can be pinned by a single edge.—and future graph densification may discover a defensible relation only if it preserves that boundary.

This entry under conditions is a kind of Graph Invariant.

Approved unparented node. No current live node supplies a defensible necessary genus or structural prerequisite for Width of a hypergraph. The reviewed identity is: In graph theory, there are two related properties of a hypergraph that are called its "width". The accelerated suggestion was declined because topical or lexical similarity does not establish hierarchy; the node is admitted without a parent pending later graph densification.
Related reasoning operations. Evidence, representation, comparison, classification, transformation, or evaluation may participate in particular cases, but participation does not make any one of them a necessary parent of every instance.

Relationships to Other Abstractions¶

Current abstraction Width of a hypergraph Domain-specific

Parents (1) — more general patterns this builds on

Width of a hypergraph is a kind of, conditional Graph Invariant Domain-specific

Supported only after specifying which width definition and hypergraph category is intended; the live name covers two related parameters.

Condition / exception Supported only after specifying which width definition and hypergraph category is intended; the live name covers two related parameters.

Hierarchy path (1) — routes to 1 parentless root

Width of a hypergraph → Graph Invariant

Neighborhood in Abstraction Space¶

Width of a hypergraph sits in a moderately populated region (47^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Combinatorial Optimization & Discrete Structures (31 abstractions)

Nearest neighbors

Strip packing problem — 0.87
Filling radius — 0.87
S-procedure — 0.87
Julia set — 0.86
Rooted product of graphs — 0.86

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Classification. The parent omits the specialist differentia. Tell: Can the case establish In graph theory, there are two related properties of a hypergraph that are called its "width"?
Line graph of a hypergraph. The graph whose vertices are a hypergraph’s hyperedges and whose adjacency records nonempty intersection between the corresponding hyperedges. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Baranyai's theorem. The theorem that a complete uniform hypergraph can be decomposed into perfect matchings whenever the edge size divides the number of vertices. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Intersection number (graph theory). The minimum ground-set size needed to represent a graph as intersections among finite vertex-associated sets, equivalently its minimum edge-clique cover size. Tell: Which entry's carrier, operation, and failure condition are satisfied?
A measurement, proxy, or consequence. Those may provide evidence without being the identity. Tell: Would Width of a hypergraph remain present if the detector or downstream effect changed?
A metaphorical analogue. A similar shape outside mathematics_logic_statistics lacks the specialist mechanism. Tell: Do the native roles transfer literally, or only the parent Classification?

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Width_of_a_hypergraph (revision 1292477178).
Preserved source candidate: https://onlinelibrary.wiley.com/doi/abs/10.1002/1097-0118%28200010%2935%3A2%3C83%3A%3AAID-JGT2%3E3.0.CO%3B2-V
Preserved source candidate: https://doi.org/10.1007/s004930170006
Preserved source candidate: https://doi.org/10.1007/s004930170001
Preserved source candidate: https://doi.org/10.1007/s00493-007-2086-y

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.