Pathwidth¶
Measure how narrowly a graph can be arranged along a path by minimizing the largest bag size minus one over path decompositions with vertex continuity and edge coverage.
Core Idea¶
The pathwidth of a finite undirected graph \(G\) is the minimum, over all path decompositions \((B_1,\ldots,B_m)\), of \(\max_i |B_i|-1\). A path decomposition is a sequence of vertex bags such that every vertex occurs in at least one bag, every edge has both endpoints together in some bag, and for each vertex the indices of bags containing it form a contiguous interval. The subtraction of one is conventional and makes a path with at least one edge have pathwidth one. Empty-graph and isolated-vertex conventions should be stated when they matter.
Scope of Application¶
The abstraction is literal wherever practitioners can identify the same constitutive roles, apply the same boundary tests, and obtain the same kind of output. The following habitats are uses of Pathwidth itself, not metaphors based only on resemblance.
- Structural graph theory. Classifying graphs by closeness to a thickened linear structure.
- Dynamic programming. Processing graph problems along bags while storing only boundary states.
- Parameterized complexity. Treating pathwidth as a structural parameter for fixed-parameter algorithms.
- Graph searching. Relating a minimum search frontier to decomposition width under stated conventions.
- Layout problems. Using vertex-separation formulations to reason about linear arrangements.
- Minor theory. Studying obstruction sets and forest-minor characterizations for bounded pathwidth.
Clarity¶
A clear account of Pathwidth must preserve the recognition invariant stated in the Core Idea rather than rely on the title alone. State all three decomposition conditions and the bag-size-minus-one width convention. Distinguish the width of one supplied decomposition from the graph's minimum pathwidth. Name graph class assumptions, including finite, undirected, simple, or multigraph conventions. When translating to vertex separation or searching, reconcile offsets and monotonicity rules explicitly.
Manages Complexity¶
Pathwidth manages complexity by replacing a diffuse field of observations or possible operations with a bounded role structure: input graph supplies a finite undirected graph supplies vertices and adjacency obligations.; index path supplies a linear order of bag positions restricts the decomposition's global shape.; vertex bags supplies subsets of vertices record the currently retained interface.; vertex coverage supplies every graph vertex appears somewhere in the sequence.; edge coverage supplies every edge has a bag containing both endpoints..
Abstract Reasoning¶
- Verify that every vertex appears in the proposed bag sequence. 2. Check that each edge has a common bag rather than merely adjacent endpoint bags. 3. For every vertex, verify that its bag indices form an interval. 4. Compute the maximum bag size and subtract one under the declared convention. 5. Construct lower bounds using separators, minors, brambles, or equivalent layout obligations. 6. Construct upper bounds by exhibiting a valid decomposition and simplifying unnecessary bags.
Knowledge Transfer¶
The strict upward abstraction is Decomposition. Pathwidth instantiates Decomposition because it represents a graph through overlapping components that preserve vertex and edge information, with a path-shaped overlap discipline and an optimized bottleneck. Within graph width parameters, the full mechanism transfers literally when the same roles and boundary tests recur. Beyond that domain, only the parent-level skeleton should travel. Reusing the label Pathwidth after removing its constitutive vocabulary would hide a change of mechanism behind an analogy. The honest transfer rule is therefore two-stage: recognize the domain-specific pattern first, then lift only the parent relation that remains invariant under a substrate change.
Relationships to Other Abstractions¶
Current abstraction Pathwidth Domain-specific
Parents (1) — more general patterns this builds on
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Pathwidth is a kind of Decomposition Prime
Pathwidth instantiates Decomposition because it represents a graph through overlapping components that preserve vertex and edge information, with a path-shaped overlap discipline and an optimized bottleneck.
Hierarchy path (1) — routes to 1 parentless root
- Pathwidth → Decomposition
Neighborhood in Abstraction Space¶
Pathwidth sits in a sparse region of the domain-specific corpus (73rd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Graph Density, Distance & Planarity (11 abstractions)
Nearest neighbors
- Subgraph Isomorphism Problem — 0.84
- Partial k-tree — 0.84
- Treewidth — 0.84
- Graph Sphericity — 0.83
- Simplex Graph — 0.83
Computed from structural-signature embeddings · 2026-09-08