Directed graph

Relatedconcepts

Isolatedpoint:ThepointinVthatisnotassociatedwithanyedgeinEiscalledtheisolatedpointofD.

Simplegraph:Agraphwithoutparalleledgesiscalledasimplegraph.

Completegraph:BetweenanytwoverticesUanduinthegraph,thereareexactlytwodirectededges(u,v),and(v,u),thenCallthedirectedgraphDacompletegraph.

Basicgraph:RemoveeachedgeofthedirectedgraphDtogetacorrespondingundirectedgraphG,whichiscalledthebasicgraphofD.CallDthedirectionalgraphofG.

Stronglyconnectedgraph:GivenadirectedgraphG=(VE),andgivenanytwonodesuandvinthegraphG,ifthenodeuItismutuallyreachablewithnodev,thatis,thereisatleastonepaththatcanstartfromnodeuandendatnodev,andthereisatleastonepaththatcanstartfromnodevandendatnodeu,thenitissaidthatthereshouldbeThedigraphGisastronglyconnectedgraph.

Weaklyconnectedgraph:Ifatleastonepairofnodesdoesnotsatisfyone-wayconnectivity,butafterremovingtheedgedirection,itisaconnectedgraphfromthepointofviewofanundirectedgraph,thenDiscalledItisaweaklyconnectedgraph.

One-wayconnectedgraph:Ifeachpairofnodesisconnectedinatleastonedirection,thenDiscalledaone-wayconnectedgraph.

Stronglyconnectedcomponent:TheextremelystronglyconnectedsubgraphofthedirectedgraphGiscalledthestronglyconnectedcomponentofthedirectedgraph.

Directedpath:Thereisalwayssuchanindependentset5inacyclicdirectedgraphD,sothatanypointiny-Js",thereexistsH∈S,fromMto"Thereisadirectedpathwithalengthnotexceeding2.

Adjacencymatrix

Exceptforisolatedvertices,anyvertexisassociatedwithatleastoneedge.Therefore,anydirectedgraph,regardlessofisolatedvertices,canbecompletelydescribedbyitsedgeset.Forexample,iftheedgesofDareasfollows:

(1,1),(1,2),(1,3),(1,4),(2,2),(2,3),(2,4),(3,3),(3,4),(4,4),

NotethatwelisttheedgesofDaccordingtothedictionarysequence,buthereisnota,B,c,...,but1,2,3....

Accordingtothisidea,wecanuseamatrixtocompletelydescribeanydirectedgraph.ThisisthedirectedgraphAdjacencymatrix.

Solvingtheshortestpath

Fortheshortestpathproblemofadirectedgraph,thecalculationstepsarethesameassolvingtheshortestpathproblemofanundirectedgraph.ThemaindifferenceisthattheshortestpathproblemofanundirectedgraphusesasingleLabelingmethod.Thesingle-labelingmethodistoassignaright-of-waylabeltoeachpoint;andforthedirectedshortestpathproblem,thedouble-labelingmethodisused.Thedoublelabelingmethodistoassigntwolabelstoeachpoint:thepathandtherightofway.

Reachability

Foranundirectedgraph,ifitisconnected,thentheremustbeapathbetweenanytwoverticesofit.Therefore,throughthisApathcan"reach"fromonevertextoanothervertex.Ifthevertex"canreachu,thenitcanalsoreachu",thatis,vanduaremutuallyreachable.

Fordirectedgraphs,thesituationisdifferent,becausethereisapathfromutov,whichdoesnotimplythatthereisalsoapathfromvtou.

SupposeDisadirectedgraph,andu,v∈D,ifthereisapathfromvertexutovertexv,thenitissaidthatvertexvtovertexuisreachable.

Theconceptofreachabilityhasnothingtodowiththenumberandlengthofthevariouspathsfromutov.Inaddition,forthesakeofcompleteness,itisstipulatedthatanyvertextoitselfisreachable.

Accessibilityisabinaryrelationshipbetweentheverticesofadirectedgraph.Accordingtothedefinition,itisreflexiveandtransitive.Generallyspeaking,reachabilityisneithersymmetricnorantisymmetric.

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