Width of a hypergraph¶
In graph theory, there are two related properties of a hypergraph that are called its "width".
Core Idea¶
Width of a hypergraph is treated here as the recurring mathematicslogicstatistics 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.
Scope of Application¶
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Documented setting. The width of a hypergraph is used in Hall-type theorems for hypergraphs.
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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.
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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.
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Examples. mw(H) = 1, since every matching can be pinned by a single edge.
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Examples. There are two matchings: is pinned e.g. by { {A,b} }, and { {A,b}, {B,a} } is pinned e.g. by { {A, a} }.
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".
Manages Complexity¶
Width of a hypergraph compresses multiple mathematicslogicstatistics 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.
Abstract Reasoning¶
- Type the carrier. Identify the mathematicslogicstatistics 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.
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.
Relationships to Other Abstractions¶
Current abstraction Width of a hypergraph Domain-specific
Parents (1) — more general patterns this builds on
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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 (47th 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