Claw-free graph¶

A graph containing no induced subgraph isomorphic to the four-vertex star K1,3.

Version: v1 · 2026-09-08 · History
Domain-specific #: 3706
Origin domain: graph theory
Subdomain: forbidden induced subgraphs

Core Idea¶

A claw-free graph excludes the claw K1,3 as an induced subgraph. For every vertex, no three of its neighbors can be mutually nonadjacent, preventing one center from inducing exactly three independent leaves. 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.

The load-bearing residual is not the broad topic of graph theory. It is hereditary graph class defined by one forbidden induced star. That residual remains recognizable when examples, notation, scale, or implementation change, but it disappears if the carrier is mistyped, the condition that no vertex subset induces precisely a K1,3 under the graph's adjacency relation fails, a neighboring object is substituted, or notation and topical resemblance replace the constitutive test.

Scope of Application¶

Claw-free graph belongs to graph theory and is useful where the analyst can specify a simple graph G, four-vertex subsets, induced subgraphs, the star K1,3 with one center and three pairwise nonadjacent leaves, vertex neighborhoods, line-graph examples and hereditary graph class, then evaluate no vertex subset induces precisely a K1,3 under the graph's adjacency relation. The scope is broad within that domain but bounded by the need for no vertex subset induces precisely a K1,3 under the graph's adjacency relation. 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 no vertex subset induces precisely a K1,3 under the graph's adjacency relation 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 Claw-free graph 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 Claw-free graph. Claw-free 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: a simple graph G, four-vertex subsets, induced subgraphs, the star K1,3 with one center and three pairwise nonadjacent leaves, vertex neighborhoods, line-graph examples and hereditary graph class. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express no vertex subset induces precisely a K1,3 under the graph's adjacency relation independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of graph theory because they reuse a simple graph G, four-vertex subsets, induced subgraphs, the star K1,3 with one center and three pairwise nonadjacent leaves, vertex neighborhoods, line-graph examples and hereditary graph class, For every vertex, no three of its neighbors can be mutually nonadjacent, preventing one center from inducing exactly three independent leaves., and type the carrier, state every parameter and convention in the definition, test that no vertex subset induces precisely a K1,3 under the graph's adjacency relation, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Claw-free graph Domain-specific

Parents (1) — more general patterns this builds on

Claw-free graph is a kind of Constraint Prime

The proposed strict upward parent is prime:constraint.

Hierarchy path (1) — routes to 1 parentless root

Claw-free graph → Constraint

Neighborhood in Abstraction Space¶

Claw-free graph sits in a crowded region of the domain-specific corpus (9^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