Contact graph¶
Represent a family of geometric objects by one vertex per object and an edge exactly when two objects satisfy a declared boundary-contact relation without prohibited interior overlap or crossing, making the permitted notion of touching part of the graph class.
Core Idea¶
A contact graph has one vertex for each object in a geometric family and an edge between two vertices when the corresponding objects touch according to a specified contact rule, ordinarily without crossing or overlapping interiors. geometry induces adjacency: the object representation fixes which pairs meet at allowed boundary points or segments, and forgetting coordinates while retaining those adjacencies yields an abstract graph whose realizability depends on object shape and contact constraints.
Its autonomous residual is the graph induced by a restrictive geometric touching relation and the existence of a compatible object representation, rather than unrestricted set intersection, proximity, graph drawing, or a graph product.
Scope of Application¶
Contact graph applies when the analyst can specify a finite or explicitly bounded family of geometric objects embedded in an ambient space, together with a typed notion of admissible contact and establish that there is a declared object-to-vertex bijection and edge equivalence with admissible geometric contact, with interior overlap, crossing, containment, higher-order meetings, and degeneracies handled by explicit convention. The entry treats graph representations and realizability; physical contact mechanics, force transmission, friction, and tolerance engineering require additional models not contained in adjacency alone.
Clarity¶
A clear claim names the carrier, governing rule, assumptions, and recognition test. This matters because touching can mean point tangency, shared positive-length boundary, closed-set intersection, or tolerance-based proximity, and different conventions define different graph classes. The disciplined statement is that the object counts as Contact graph exactly when there is a declared object-to-vertex bijection and edge equivalence with admissible geometric contact, with interior overlap, crossing, containment, higher-order meetings, and degeneracies handled by explicit convention
Manages Complexity¶
The abstraction compresses circle and disk contacts, penny graphs, smooth convex-body packings, rectangle and square contacts, polygon, segment, arc, triangle, three-dimensional body contacts, weighted contacts, and degenerate or generic-position conventions into a stable carrier, rule, invariant, and failure boundary. It makes comparison tractable while retaining the variables that control validity.
Abstract Reasoning¶
- Type the carrier. Establish a finite or explicitly bounded family of geometric objects embedded in an ambient space, together with a typed notion of admissible contact and reject examples from a different problem. 2. Lock the rule. Express that there is a declared object-to-vertex bijection and edge equivalence with admissible geometric contact, with interior overlap, crossing, containment, higher-order meetings, and degeneracies handled by explicit convention independently of one notation or implementation.
Knowledge Transfer¶
Transfer within mathematics is strong when new cases preserve the same carrier, mechanism, and diagnostic. The move from The circle-packing theorem represents every finite planar graph by disks with disjoint interiors whose tangencies encode its edges. to A floorplanning abstraction represents interior-disjoint rectangles as vertices and shared boundary segments as adjacencies. demonstrates that continuity.
Outside the domain, only the skeleton—replace spatial objects by nodes and retain exactly the adjacency created at their permitted boundaries while discarding most geometric detail—travels automatically.
Relationships to Other Abstractions¶
Current abstraction Contact graph Domain-specific
Parents (1) — more general patterns this builds on
-
Contact graph is a kind of Network Prime
The proposed strict upward parent is
prime:network.
Hierarchy path (1) — routes to 1 parentless root
- Contact graph → Network → Reservoir-Flux Network → Conservation Laws → Invariance
Neighborhood in Abstraction Space¶
Contact graph sits in a crowded region of the domain-specific corpus (37th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Convex Geometry & Spatial Partition (35 abstractions)
Nearest neighbors
- Map graph — 0.91
- Polygon partition — 0.90
- Ambient space (mathematics) — 0.90
- Triangulation (topology) — 0.89
- Ran space — 0.89
Computed from structural-signature embeddings · 2026-09-08