Clique-sum¶
Combine graphs by choosing equally sized complete subgraphs, identifying their vertices through a bijection, and optionally deleting interface-clique edges under an explicit convention.
Core Idea¶
A clique-sum is a graph-composition operation that takes the disjoint union of two graphs and identifies corresponding vertices of selected equal-order cliques, with interface-edge deletion governed by the chosen convention. Disjoint summands preserve their off-interface vertices and edges, quotient identification fuses matched clique vertices, and the edge convention determines which clique edges survive in the resulting graph The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.
Scope of Application¶
Clique-sum belongs to graph theory and is useful where the analyst can specify two finite graphs, a clique in each, and a bijection between the selected clique vertices, then evaluate the overlap is exactly the image of two complete subgraphs matched by a declared bijection, and all changes outside that interface are excluded. The scope is broad within that domain but bounded by the need for the overlap is exactly the image of two complete subgraphs matched by a declared bijection, and all changes outside that interface are excluded. The entry treats finite undirected graph conventions; matroid sums and topological connected sums are related analogies but require their own definitions.
Clarity¶
The abstraction clarifies a crowded vocabulary by making the overlap is exactly the image of two complete subgraphs matched by a declared bijection, and all changes outside that interface are excluded the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because some sources reserve k-sum for exact order and others for order at most k, while edge-deletion rules range from none to all.
Manages Complexity¶
Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Clique-sum. Clique-sum compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.
Abstract Reasoning¶
- Identify the carrier. State what the elements, states, objects, or observations are: two finite graphs, a clique in each, and a bijection between the selected clique vertices. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the overlap is exactly the image of two complete subgraphs matched by a declared bijection, and all changes outside that interface are excluded independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of graph theory because they reuse two finite graphs, a clique in each, and a bijection between the selected clique vertices, Disjoint summands preserve their off-interface vertices and edges, quotient identification fuses matched clique vertices, and the edge convention determines which clique edges survive in the resulting graph, and verify each interface is a clique of the stated order, record whether order means exactly or at most \(k\), state the vertex bijection and edge-deletion convention, and reconstruct each summand away from the shared interface.
Relationships to Other Abstractions¶
Current abstraction Clique-sum Domain-specific
Parents (1) — more general patterns this builds on
-
Clique-sum is a kind of Transformation Prime
The proposed strict upward parent is
prime:transformation.
Hierarchy path (1) — routes to 1 parentless root
- Clique-sum → Transformation → Function (Mapping)
Neighborhood in Abstraction Space¶
Clique-sum sits in a crowded region of the domain-specific corpus (28th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Unclustered & Miscellaneous (1565 abstractions)
Nearest neighbors
- Split graph — 0.91
- Biclique-free graph — 0.91
- Intersection number (graph theory) — 0.91
- Clique graph — 0.91
- Join (graph theory) — 0.91
Computed from structural-signature embeddings · 2026-09-08