Thickness (graph theory)¶

The minimum number of planar spanning subgraphs needed to partition a graph's edge set.

Version: v1 · 2026-09-08 · History
Domain-specific #: 7128
Origin domain: topological graph theory
Subdomain: topological graph theory

Core Idea¶

Graph thickness measures how many planar layers are required to draw all edges without crossings within a layer while sharing the original vertex set. An edge coloring assigns each edge to a layer, planarity is tested separately in every color class, and optimization minimizes the number of nonempty layers. 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¶

Thickness (graph theory) belongs to topological graph theory and is useful where the analyst can specify the typed topological graph theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate all original vertices are retained, every edge belongs to exactly one layer, every layer is planar under the declared graph convention, and no decomposition uses fewer layers. The scope is broad within that domain but bounded by the need for all original vertices are retained, every edge belongs to exactly one layer, every layer is planar under the declared graph convention, and no decomposition uses fewer layers. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.

Clarity¶

The abstraction clarifies a crowded vocabulary by making all original vertices are retained, every edge belongs to exactly one layer, every layer is planar under the declared graph convention, and no decomposition uses fewer layers 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 Thickness (graph theory). Thickness (graph theory) 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 topological 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. Lock the constitutive rule. Express all original vertices are retained, every edge belongs to exactly one layer, every layer is planar under the declared graph convention, and no decomposition uses fewer layers independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of topological graph theory because they reuse the typed topological graph theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, An edge coloring assigns each edge to a layer, planarity is tested separately in every color class, and optimization minimizes the number of nonempty layers., and type the carrier, state every parameter and convention in the definition, test that all original vertices are retained, every edge belongs to exactly one layer, every layer is planar under the declared graph convention, and no decomposition uses fewer layers, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Thickness (graph theory) Domain-specific

Parents (1) — more general patterns this builds on

Thickness (graph theory) is a kind of Partition Prime

The proposed strict upward parent is prime:partition.

Hierarchy path (1) — routes to 1 parentless root

Thickness (graph theory) → Partition → Set and Membership

Neighborhood in Abstraction Space¶

Thickness (graph theory) sits in a crowded region of the domain-specific corpus (1^st 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

Computed from structural-signature embeddings · 2026-09-08