Double graph¶
A graph construction that replaces every vertex with two nonadjacent copies and replaces each original edge with all four edges between the corresponding copy pairs.
Core Idea¶
The double graph is expressible as a lexicographic or direct product under stated loop conventions and systematically doubles vertices and quadruples edges while preserving interpretable structural relations. Two layers of the original vertex set are formed; each original adjacency induces within-layer and cross-layer adjacencies between every copy of its endpoints, while copies of the same vertex remain nonadjacent. 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¶
Double graph belongs to graph products and transformations and is useful where the analyst can specify the typed graph products and transformations carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate the input is a simple graph, copy labels, product convention and looped factor, induced edge rule, absence of same-vertex copy edges, and resulting vertex and edge counts are explicit. The scope is broad within that domain but bounded by the need for the input is a simple graph, copy labels, product convention and looped factor, induced edge rule, absence of same-vertex copy edges, and resulting vertex and edge counts are explicit.
Clarity¶
The abstraction clarifies a crowded vocabulary by making the input is a simple graph, copy labels, product convention and looped factor, induced edge rule, absence of same-vertex copy edges, and resulting vertex and edge counts are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test.
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 Double graph. Double graph 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: the typed graph products and transformations carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the input is a simple graph, copy labels, product convention and looped factor, induced edge rule, absence of same-vertex copy edges, and resulting vertex and edge counts are explicit independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of graph products and transformations because they reuse the typed graph products and transformations carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, Two layers of the original vertex set are formed; each original adjacency induces within-layer and cross-layer adjacencies between every copy of its endpoints, while copies of the same vertex remain nonadjacent., and type the carrier, state every parameter and convention in the definition, test that the input is a simple graph, copy labels, product convention and looped factor, induced edge rule, absence of same-vertex copy edges, and resulting vertex and edge counts are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Double graph Domain-specific
Parents (1) — more general patterns this builds on
-
Double graph is a kind of Transformation Prime
The proposed strict upward parent is
prime:transformation.
Hierarchy path (1) — routes to 1 parentless root
- Double graph → Transformation → Function (Mapping)
Neighborhood in Abstraction Space¶
Double graph sits in a crowded region of the domain-specific corpus (2nd percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Graph Invariants & Constructions (49 abstractions)
Nearest neighbors
- Join (graph theory) — 0.95
- Split graph — 0.95
- Self-complementary graph — 0.94
- Strong product of graphs — 0.94
- Bivariegated graph — 0.94
Computed from structural-signature embeddings · 2026-09-08