Claw-free graph¶

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

Version: v2 · 2026-09-19 · History
Domain-specific #: 3706
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Graph Theory, Forbidden Induced Subgraphs → Mathematics

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.

How would you explain it like I'm…

No Three Stranger Friends

Draw dots for kids and lines between kids who are friends. The picture is claw-free if no kid has three friends who are all strangers to each other. If any kid has three friends, at least two of those friends must know each other.

The No-Claw Rule

A graph is a drawing of dots (vertices) joined by lines (edges). A claw is one center dot joined to three other dots, where those three have no lines between them, so it looks like a claw or a three-pronged star. A graph is claw-free if you can never pick four dots that form exactly this shape, with no extra lines among the three outer dots. So for every dot, among any three of its neighbors, at least two are joined to each other.

Forbidden Induced Star

A claw is the graph K1,3: one center vertex adjacent to three leaves, with no edges among the leaves. A graph is claw-free if it contains no claw as an induced subgraph, meaning no set of four vertices has exactly the claw's edges and no others. Equivalently, for every vertex, no three of its neighbors are pairwise nonadjacent. The word 'induced' matters: a claw-free graph can still contain a claw as an ordinary subgraph if extra edges among the leaves are present. Because deleting vertices cannot create an induced claw, claw-free graphs form a hereditary class.

A claw-free graph is one that excludes the claw K1,3 as an induced subgraph. Concretely, no vertex has three pairwise nonadjacent neighbors; equivalently, the independence number of every open neighborhood is at most two. The definition is about induced subgraphs, not ordinary subgraphs: a vertex with three neighbors is fine as long as at least one pair of them is adjacent. Because the condition is a single forbidden induced subgraph, the class is hereditary, closed under taking induced subgraphs. Testing membership is a local check over neighborhoods under the graph's adjacency relation. Confusing it with 'no vertex of degree 3' or with forbidding K1,3 as a (non-induced) subgraph describes a different, much smaller class.

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.
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) — outgoing typed relations

Claw-free graph is a kind of Constraint Prime

Membership is a structural constraint excluding a particular induced configuration.

Hierarchy path (1) — routes to 1 parentless root

Claw-free graph → is a kind of Constraint

Neighborhood in Abstraction Space¶

Claw-free graph sits in a crowded region of the domain-specific corpus (5^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Graph Classes & Invariants (40 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25