Bidimensionality¶
A graph-algorithm framework for parameters that grow quadratically on grid-like graphs and do not increase under minors or contractions, enabling subexponential and kernel results.
Core Idea¶
Bidimensionality theory combines graph-minor structure with a parameter's monotonicity and grid growth to obtain treewidth bounds, approximation schemes, kernels, and fixed-parameter algorithms on sparse graph classes. A large parameter-obstruction grid forces large value; conversely a small parameter excludes large grids and bounds treewidth, after which dynamic programming or decomposition algorithms apply. 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¶
Bidimensionality belongs to parameterized graph algorithms and is useful where the analyst can specify the typed parameterized graph algorithms carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate the graph class, parameterized problem, minor- or contraction-monotonicity, quadratic grid-growth condition, and algorithmic meta-theorem hypotheses are all satisfied. The scope is broad within that domain but bounded by the need for the graph class, parameterized problem, minor- or contraction-monotonicity, quadratic grid-growth condition, and algorithmic meta-theorem hypotheses are all satisfied. 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 the graph class, parameterized problem, minor- or contraction-monotonicity, quadratic grid-growth condition, and algorithmic meta-theorem hypotheses are all satisfied 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 Bidimensionality 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 Bidimensionality. Bidimensionality 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 parameterized graph algorithms 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 graph class, parameterized problem, minor- or contraction-monotonicity, quadratic grid-growth condition, and algorithmic meta-theorem hypotheses are all satisfied independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of parameterized graph algorithms because they reuse the typed parameterized graph algorithms carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, A large parameter-obstruction grid forces large value; conversely a small parameter excludes large grids and bounds treewidth, after which dynamic programming or decomposition algorithms apply., and type the carrier, state every parameter and convention in the definition, test that the graph class, parameterized problem, minor- or contraction-monotonicity, quadratic grid-growth condition, and algorithmic meta-theorem hypotheses are all satisfied, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Bidimensionality Domain-specific
Parents (1) — more general patterns this builds on
-
Bidimensionality is a kind of Classification Prime
The proposed strict upward parent is
prime:classification.
Hierarchy path (1) — routes to 1 parentless root
- Bidimensionality → Classification
Neighborhood in Abstraction Space¶
Bidimensionality sits in a crowded region of the domain-specific corpus (14th 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
- Planarity testing — 0.92
- Dissociation number — 0.92
- Treewidth — 0.92
- Split graph — 0.91
- Matching (graph theory) — 0.91
Computed from structural-signature embeddings · 2026-09-08