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<Chapter Label="Homomorphisms"><Heading>Homomorphisms</Heading>
<!--Let $E_a=(E_a^0,E_a^1,\ra_a,\so_a)$ and $E_b=(E_b^0,E_b^1,\ra_b,\so_b)$ be two graphs, and let $\phi_0 : E_a^0 \to E_b^0$ and $\phi_1 : E_a^1 \to E_b^1$ be functions. Then the pair $\phi = (\phi_0, \phi_1)$ is a \emph{graph homomorphism from} $E_a$ \emph{to} $E_b$ if $\phi_0(\so_a(e)) = \so_b(\phi_1(e))$ and $\phi_0(\ra_a(e)) = \ra_b(\phi_1(e))$ for every $e \in E_a^1$. If $\phi_0$ and $\phi_1$ are in addition bijective, then $\phi$ is a \emph{graph isomorphism from} $E_a$ \emph{to} $E_b$. In this case we say that $E_a$ and $E_b$ are \emph{isomorphic} and write $E_a \cong E_b$.--> <Section><Heading>Acting on digraphs</Heading> <#Include Label="OnDigraphs"> <#Include Label="OnMultiDigraphs"> <#Include Label="OnTuplesDigraphs"> </Section> <Section Label="Isomorphisms and canonical labellings"> <Heading>Isomorphisms and canonical labellings</Heading> From version 0.11.0 of &Digraphs; it is possible to use either &BLISS; or &NAUTY; (via &NautyTracesInterface;) to calculate canonical labellings and automorphism groups of digraphs; see <Cite Key="JK07"/> and <Cite Key="MP14"/> for more details about &BLISS; and &NAUTY;, respectively. <#Include Label="DigraphsUseNauty"> <#Include Label="AutomorphismGroupDigraph"> <#Include Label="BlissAutomorphismGroup"> <#Include Label="NautyAutomorphismGroup"> <#Include Label="AutomorphismGroupDigraphColours"> <#Include Label="AutomorphismGroupDigraphEdgeColours"> <#Include Label="BlissCanonicalLabelling"> <#Include Label="BlissCanonicalLabellingColours"> <#Include Label="BlissCanonicalDigraph"> <#Include Label="DigraphGroup"> <#Include Label="DigraphOrbits"> <#Include Label="DigraphOrbitReps"> <#Include Label="DigraphSchreierVector"> <#Include Label="DigraphStabilizer"> <#Include Label="IsIsomorphicDigraph"> <#Include Label="IsIsomorphicDigraphColours"> <#Include Label="IsomorphismDigraphs"> <#Include Label="IsomorphismDigraphsColours"> <#Include Label="RepresentativeOutNeighbours"> <#Include Label="IsDigraphAutomorphism"> <#Include Label="IsDigraphColouring"> <#Include Label="MaximalCommonSubdigraph"> <#Include Label="MinimalCommonSuperdigraph"> </Section> <Section><Heading>Homomorphisms of digraphs</Heading> The following methods exist to find homomorphisms between digraphs. If an argument to one of these methods is a digraph with multiple edges, then the multiplicity of edges will be ignored in order to perform the calculation; the digraph will be treated as if it has no multiple edges. <#Include Label="HomomorphismDigraphsFinder"> <#Include Label="DigraphHomomorphism"> <#Include Label="HomomorphismsDigraphs"> <#Include Label="DigraphMonomorphism"> <#Include Label="MonomorphismsDigraphs"> <#Include Label="DigraphEpimorphism"> <#Include Label="EpimorphismsDigraphs"> <#Include Label="DigraphEmbedding"> <#Include Label="EmbeddingsDigraphs"> <#Include Label="IsDigraphHomomorphism"> <#Include Label="IsDigraphEmbedding"> <#Include Label="SubdigraphsMonomorphisms"> <#Include Label="DigraphsRespectsColouring"> <#Include Label="GeneratorsOfEndomorphismMonoid"> <#Include Label="DigraphColouring"> <#Include Label="DigraphGreedyColouring"> <#Include Label="DigraphWelshPowellOrder"> <#Include Label="ChromaticNumber"> <#Include Label="DigraphCore"> <#Include Label="LatticeDigraphEmbedding"> <#Include Label="IsLatticeHomomorphism"> </Section> </Chapter> ¤ Dauer der Verarbeitung: 0.17 Sekunden (vorverarbeitet) ¤*© Formatika GbR, Deutschland |
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2026-03-28