Map graph¶
The intersection graph of finitely many interior-disjoint simply connected regions in the plane, with vertices for regions and adjacency whenever two regions meet at any boundary point.
Core Idea¶
Map graphs include planar graphs but allow arbitrarily large cliques when many regions meet at one point; they are equivalently half-squares of suitable planar bipartite graphs under standard representations. A planar map supplies one graph vertex per region, every point or boundary contact shared by regions generates pairwise adjacency, and a witness bipartite graph connects regions to their shared contact points. 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¶
Map graph belongs to structural and geometric graph theory and is useful where the analyst can specify the typed structural and geometric graph theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate the finite region family, Euclidean plane and simple-connectivity convention, pairwise interior disjointness, boundary-contact adjacency including point contacts, graph simplicity, planar witness bipartite graph and half-square construction, clique behavior, recognition representation and comparison with planar and intersection graphs are explicit.
Clarity¶
The abstraction clarifies a crowded vocabulary by making the finite region family, Euclidean plane and simple-connectivity convention, pairwise interior disjointness, boundary-contact adjacency including point contacts, graph simplicity, planar witness bipartite graph and half-square construction, clique behavior, recognition representation and comparison with planar and intersection graphs 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 Map graph. Map 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 structural and geometric graph theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of structural and geometric graph theory because they reuse the typed structural and geometric graph theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, A planar map supplies one graph vertex per region, every point or boundary contact shared by regions generates pairwise adjacency, and a witness bipartite graph connects regions to their shared contact points., and type the carrier, state every parameter and convention in the definition, test that the finite region family, Euclidean plane and simple-connectivity convention, pairwise interior disjointness, boundary-contact adjacency including point contacts, graph simplicity, planar witness bipartite graph and half-square construction, clique behavior, recognition representation and comparison with planar and intersection graphs are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Map graph Domain-specific
Parents (1) — more general patterns this builds on
-
Map graph is a kind of Intersection Prime
The proposed strict upward parent is
prime:intersection.
Hierarchy path (1) — routes to 1 parentless root
- Map graph → Intersection → Set and Membership
Neighborhood in Abstraction Space¶
Map 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
- Graph isomorphism — 0.95
- Split graph — 0.94
- Join (graph theory) — 0.94
- Orientation (graph theory) — 0.93
- Biregular graph — 0.93
Computed from structural-signature embeddings · 2026-09-08