Rank-width¶

A graph width parameter that minimizes, over subcubic decomposition trees, the maximum binary rank of the adjacency matrix crossing any induced vertex cut.

Version: v1 · 2026-09-08 · History
Domain-specific #: 6396
Origin domain: graph theory
Subdomain: width parameters and decompositions

Core Idea¶

Rank-width is the minimum across rank decompositions of the maximum GF(2) rank of the bipartite adjacency matrix between the two vertex sides induced by any decomposition-tree edge. Each tree edge tests how many linearly independent neighborhood patterns cross its cut; a decomposition arranges vertices to keep the worst cut rank small, and the optimum gives the graph parameter. 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¶

Rank-width belongs to graph theory and is useful where the analyst can specify a graph, a subcubic tree whose leaves are graph vertices, edge-induced bipartitions, cross-adjacency matrices over GF(2), cut ranks and a minimax width, then evaluate the decomposition leaves correspond bijectively to graph vertices, each cut matrix uses adjacency over GF(2), width is the maximum cut rank and rank-width minimizes that maximum. The scope is broad within that domain but bounded by the need for the decomposition leaves correspond bijectively to graph vertices, each cut matrix uses adjacency over GF(2), width is the maximum cut rank and rank-width minimizes that maximum.

Clarity¶

The abstraction clarifies a crowded vocabulary by making the decomposition leaves correspond bijectively to graph vertices, each cut matrix uses adjacency over GF(2), width is the maximum cut rank and rank-width minimizes that maximum 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 the name Rank-width can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.

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 Rank-width. Rank-width 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: a graph, a subcubic tree whose leaves are graph vertices, edge-induced bipartitions, cross-adjacency matrices over GF(2), cut ranks and a minimax width. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the decomposition leaves correspond bijectively to graph vertices, each cut matrix uses adjacency over GF(2), width is the maximum cut rank and rank-width minimizes that maximum independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of graph theory because they reuse a graph, a subcubic tree whose leaves are graph vertices, edge-induced bipartitions, cross-adjacency matrices over GF(2), cut ranks and a minimax width, Each tree edge tests how many linearly independent neighborhood patterns cross its cut; a decomposition arranges vertices to keep the worst cut rank small, and the optimum gives the graph parameter., and type the carrier, state every parameter and convention in the definition, test that the decomposition leaves correspond bijectively to graph vertices, each cut matrix uses adjacency over GF(2), width is the maximum cut rank and rank-width minimizes that maximum, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Rank-width Domain-specific

Parents (1) — more general patterns this builds on

Rank-width is a kind of Decomposition Prime

The proposed strict upward parent is prime:decomposition.

Hierarchy path (1) — routes to 1 parentless root

Rank-width → Decomposition

Neighborhood in Abstraction Space¶

Rank-width sits in a crowded region of the domain-specific corpus (10^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Graph Connectivity & Network Measures (31 abstractions)

Nearest neighbors

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