It plays an important role in the structure theory for claw-free graphs by .

This characterization provided the initial motivation for studying claw-free graphs.

Graphs formed in four simple ways from smaller claw-free graphs.

Perfect claw-free graphs may be recognized in polynomial time.

Every 2-vertex-connected claw-free graph with an odd number of vertices is factor-critical.

A generalization of the same technique can also be used to find maximum independent sets in claw-free graphs.

The complement of any triangle-free graph is a claw-free graph.

However, there exist claw-free graphs that are not interval graphs.

This has been independently extended to an algorithm for all claw-free graphs by and .

The graph K is called a claw, and is used to define the claw-free graphs.

It is possible to color perfect claw-free graphs, or to find maximum cliques in them, in polynomial time.

Claw-free graphs have this property.

Six specific subclasses of claw-free graphs.

If a claw-free graph is not perfect, it is NP-hard to find its largest clique.

Chudnovsky and Seymour classify arbitrary connected claw-free graphs into one of the following:

This result follows directly from the more fundamental theorem that every connected claw-free graph with an even number of vertices has a perfect matching.

Related decomposition techniques have also borne fruit in the study of other graph classes, and in particular for the claw-free graphs.

Repeatedly removing matched pairs of vertices in this way forms a perfect matching in the given claw-free graph.

It is possible to test whether a line graph, or more generally a claw-free graph, is well-covered in polynomial time.

In a perfect claw-free graph, the neighborhood of any vertex forms the complement of a bipartite graph.

Despite this domination perfectness property, it is NP-hard to determine the size of the minimum dominating set in a claw-free graph.

Although not every claw-free graph is perfect, claw-free graphs satisfy another property, related to perfection.

Claws are notable in the definition of claw-free graphs, graphs that do not have any claw as an induced subgraph.

In graph theory, an area of mathematics, a claw-free graph is a graph that does not have a claw as an induced subgraph.

All line graphs are claw-free graphs, graphs without an induced subgraph in the form of a three-leaf tree.