({{, (RefLink[LambdaComponents, paclet:ref/LambdaComponents])[g] 
gives the lambda components of the graph g.}, {, (RefLink[LambdaComponents, paclet:ref/LambdaComponents])[g,{(Subscript[v, 1]),(Subscript[v, 2]),…}]
gives the lambda components that include at least one of the vertices {(Subscript[v, 1]),(Subscript[v, 2]),…}.}, {, (RefLink[LambdaComponents, paclet:ref/LambdaComponents])[{v->w,…},…]
uses rules v->w to specify the graph g. }})

OpenersNotesSection(Spacer1)Details

(RefLink[LambdaComponents, paclet:ref/LambdaComponents]) is also known as a lambda set.

A lambda component is a set of vertices that have more edge-independent paths connecting them to each other than to outside vertices.

(RefLink[LambdaComponents, paclet:ref/LambdaComponents]) returns a list of components {(Subscript[c, 1]),(Subscript[c, 2]),…}, where each component (Subscript[c, i]) is given as a list of vertices.

(RefLink[LambdaComponents, paclet:ref/LambdaComponents])[g] is equivalent to (RefLink[LambdaComponents, paclet:ref/LambdaComponents])[g,(RefLink[VertexList, paclet:ref/VertexList])[g]].

(RefLink[LambdaComponents, paclet:ref/LambdaComponents]) works with undirected graphs, directed graphs, multigraphs, and mixed graphs.

OpenersPrimaryExamplesSection(Spacer1)Examples  (11)

OpenersExampleSection0(Spacer1)Basic Examples  (1)

Give the lambda components for a graph:

LambdaComponents[]

OpenersExampleSection1(Spacer1)Scope  (6)

(RefLink[LambdaComponents, paclet:ref/LambdaComponents]) works with undirected graphs:

LambdaComponents[]

Directed graphs:

LambdaComponents[]

Multigraphs:

LambdaComponents[]

Mixed graphs:

LambdaComponents[]

Give the components that include at least one of the vertices:

LambdaComponents[,{1,3}]

Use rules to specify the graph:

LambdaComponents[{1->2,1->3,2->4,3->4,3->5,4->6,5->6}]

OpenersExampleSection3(Spacer1)Applications  (2)

Highlight all lambda components in a graph:

g=;

Table[HighlightGraph[g,Subgraph[g,i],VertexSize->0.2],{i,LambdaComponents[g]}]

A family network consisting of the closest relatives, including parents, siblings, and children. Find a smallest family group with more than one member, where relatives have more edge-independent paths connecting them to each other than to relatives outside the group:

g=ExampleData[{"NetworkGraph","FamilyGathering"}]

The first nontrivial λ component occurs after the singletons:

λ=LambdaComponents[g];Length/@λ

HighlightGraph[g,Subgraph[g,λ[[VertexCount[g]+1]]]]

OpenersExampleSection4(Spacer1)Properties & Relations  (2)

All Luccio–Sami components are lambda components:

g=;

LuccioSamiComponents[g]

LambdaComponents[g]

MemberQ[LuccioSamiComponents[g],Alternatives @@ LambdaComponents[g]]

The converse is not always true:

MemberQ[LambdaComponents[g],{5,6,7,8}]

MemberQ[LuccioSamiComponents[g],{5,6,7,8}]

Lambda components are partially ordered by inclusion:

g=;

λ=LambdaComponents[g]

The corresponding directed acyclic graph:

AdjacencyGraph[λ,Table[Boole@(SubsetQ[λ[[j]],λ[[i]]]&&λ[[i]]!=λ[[j]]),{i,Length[λ]},{j,Length[λ]}],VertexLabels->"Name",ImagePadding->10]

AcyclicGraphQ[%]

({{DocumentationFooterIconsRelatedFunction.png, ({{See Also}, {(RefLink[LuccioSamiComponents, paclet:ref/LuccioSamiComponents])  ▪  (RefLink[ConnectedComponents, paclet:ref/ConnectedComponents])  ▪  (RefLink[KCoreComponents, paclet:ref/KCoreComponents])}})}})

({{DocumentationFooterIconsRelatedGuide.png, ({{Related Guides}, {▪(RefLinkPlain[Social Network Analysis, paclet:guide/SocialNetworks])}})}})