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.

Version: v2 · 2026-09-19 · History
Domain-specific #: 1484
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Graph Theory, Graph Decomposition and Minors → Mathematics

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.

How would you explain it like I'm…

Gluing Drawings at Matching Dots

Imagine two dot-and-line drawings. Each one has a little group of dots where every dot is joined to every other, like a triangle. You glue the two drawings together by stacking one group exactly on top of the other so those dots become the same dots. Depending on the rule you pick, you might erase some of the lines along the glued part, but nothing else changes.

Joining Graphs Along a Clique

A clique is a group of dots in a graph that are all connected to each other. A clique-sum joins two graphs by picking a clique of the same size in each one, matching their dots up one-to-one, and merging each matched pair into a single dot. Everything else in both graphs stays exactly as it was, and the two graphs only touch at that merged clique. Some versions of the rule also let you delete some lines inside the glued clique. It is a way of building bigger graphs from smaller pieces glued along clean seams.

Gluing Graphs Along Cliques

A clique-sum is an operation for composing two graphs. Take the disjoint union of the two graphs, choose a clique in each with the same number of vertices, fix a one-to-one matching between them, and identify (merge) each matched pair. Depending on the convention, some or all of the edges within the merged clique may then be deleted. Vertices and edges away from the interface are untouched, and the two pieces overlap only at the glued clique. It is different from simply finding a clique, taking a union, or joining graphs with new edges: the interface must be complete and vertices must actually be merged. Conventions differ on whether a 'k-clique-sum' means cliques of exactly k or at most k vertices, so this must be stated. Clique-sums let one describe graphs as pieces joined through small separators.

A clique-sum is a graph composition: form the disjoint union of two graphs, select a clique in each of the same order, identify them via a declared vertex bijection, and then delete interface edges as the chosen convention permits. Off-interface vertices and edges of each summand are preserved, and the overlap is exactly the image of the two matched complete subgraphs. Whether 'order k' means exactly k or at most k (a k-clique-sum) must be stated, as must the edge-deletion convention. The identity fails if an interface is not complete, if vertices are connected by new edges rather than identified, if the summands overlap outside the interface, or if deletions alter structure away from the interface. Verification checks that each interface is a clique of the stated order, records the bijection and convention, and reconstructs each summand away from the interface. Clique-sums express graphs as pieces joined through bounded clique separators, support closure proofs for graph classes, relate to treewidth and decompositions, and appear in graph-minor structure theorems.

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.
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) — outgoing typed relations

Clique-sum is a kind of Transformation Prime

Clique-sum is literally a rule-governed mapping from graphs plus interface data to a new graph, preserving off-interface structure while altering identification and selected interface edges.

Hierarchy path (1) — routes to 1 parentless root

Clique-sum → is a kind of Transformation → is a kind of, typical Function (Mapping)

Neighborhood in Abstraction Space¶

Clique-sum sits in a crowded region of the domain-specific corpus (16^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Graph Classes & Invariants (40 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 · 2026-10-25