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.