Maximum matching

In graph theory, a maximum-cardinality matching is a special kind of subgraph useful in many computational contexts.

Version

v1 · 2026-09-28 · History

Domain-specific #

10623

Domain group

Formal Sciences

Origin domain

Mathematics

Subdomain

Graph Theory → Mathematics

Core Idea

Maximum matching is treated here as the recurring mathematicslogicstatistics identity summarized by this source-grounded definition: In graph theory, a maximum-cardinality matching is a special kind of subgraph useful in many computational contexts. In graph theory, a maximum-cardinality matching is a special kind of subgraph useful in many computational contexts. Given a graph , a matching is a subgraph where no two edges share a vertex. The cardinality of the matching is the number of edges in the subgraph, and the maximum cardinality is the largest number of edges a matching can contain. graph is a maximum-cardinality.

Scope of Application

• Algorithms for arbitrary graphs. This is better in theory for sufficiently dense graphs, but in practice the algorithm is slower.

• Algorithms for bipartite graphsFlow-based algorithm. The simplest way to compute a maximum-cardinality matching is to follow the Ford–Fulkerson algorithm.

• Algorithms for bipartite graphsFlow-based algorithm. This algorithm solves the more general problem of computing the maximum flow.

• Algorithms for bipartite graphsFlow-based algorithm. A bipartite graph can be converted to a flow network as follows.

• Algorithms for bipartite graphsFlow-based algorithm. Add a source vertex ; add an edge from to each vertex in .

Clarity

A clear use of Maximum matching 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, a maximum-cardinality matching is a special kind of subgraph useful in many computational contexts.

Manages Complexity

Maximum matching compresses multiple mathematicslogicstatistics details into a stable diagnostic relation. The source shows both the central mechanism—as each path can be found in time, the running time is , and the maximum matching consists of the edges of that carry flow from to .—and the practical consequence—the same bound was achieved by an algorithm by and an algorithm by Gabow and Tarjan.

Abstract Reasoning

1. Type the carrier. Identify the mathematicslogicstatistics entities to which the claim applies.
2. State the relation. Use the source-grounded identity: In graph theory, a maximum-cardinality matching is a special kind of subgraph useful in many computational contexts.
3. Check operation and conditions. An improvement to this algorithm is given by the more elaborate Hopcroft–Karp algorithm, which searches for multiple augmenting paths simultaneously.
4. Demand recognition evidence.

Knowledge Transfer

Within the home domain. Knowledge about Maximum matching transfers literally when a new case preserves the same carrier type, relation, and recognition test. This is better in theory for sufficiently dense graphs, but in practice the algorithm is slower. The simplest way to compute a maximum-cardinality matching is to follow the Ford–Fulkerson algorithm. Beyond the home domain. No canonical parent is asserted for Maximum matching. 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.

Neighborhood in Abstraction Space

Maximum matching sits in a sparse region of the domain-specific corpus (77^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Convex Optimization & Iterative Methods (8 abstractions)

Nearest neighbors

• Convex bipartite graph — 0.89
• Matching — 0.83
• Factor-critical graph — 0.82
• A-star algorithm — 0.82
• Treewidth — 0.81

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