theory connected component cycle system edges connecting pairs cycles consecutive edges share cycles connected path consecutive edges connected names justified analogy commonly studied trees forests tree connected cycles forest union trees attribute study book linear programming arise solution certain network flow problems form models functions occur problems sparse edges relative number structure allows families sparse unions forests comes definitions structure define set edges edge coincide allow multiple edges edges pair loops edges formed edges edge edge connected component consisting edges reached following edges single given starting connected edge edge cycle connected incident exactly edges loop connected component contains cycle connected component edges components cycles trees components single cycle called trees tree connected containing exactly cycle single connected component usually called authors define tree tree tree general multiple connected components long trees trees removes tree edges cycle result tree reversing process tree connecting new edge result tree path tree connecting added edge added edge form tree unique cycle tree adding edge connects newly added result tree alternative method constructing trees start single cycle repeat operation number times edges tree unique way cycle forest tree forest contains exactly cycle certain specific types studied forest called edges added causing component contain multiple cycles contains tree components forest add edge connecting tree forming single cycle edge connecting tree component forests exactly component tree spanning need edges example edges components trees consisting single contained larger spanning conversely connected components trees forests tree component forests stated precisely consist tree component trees covering remaining directed versions definitions directed like directed consists edges edge directed directed directed outgoing edge directed forest commonly called functional directed directed exactly directed formed removing direction edge number edges set edges set exactly edges conversely property number edges induced number trees defined connected equally edges moving individual families family property family family family edges family contains instance drawn pair edges point intersection conway edges saying precise characterization true exactly cycle odd cycle conditions defining define edges exactly edges important families defined values characterized formed edge union forests sufficiently sparse random constant probability choosing uniformly random edges results tends limit large random edges large component simple self loops multiple edges number simple trees labelled formula values found sequence number directed allowing self loops possible outgoing edge fact provide proof formula number trees finding directed trees distinguished self loops allowed number directed instead functions directed sense equivalent function set interpreted defining directed edge resulting directed include self loops value alternatively self loops produces non direction directed determines function target edge goes non directed adding self loops converted function way reason directed called functional viewing function functional provides convenient language describing properties easily described function point view technique especially applicable problems involving functions correspond paths functional cycle detection problem following path functional find cycle applications number theory method finding collisions functions applications expected behave randomly study properties functional arising randomly chosen particular form birthday implies random functional path starting randomly selected typically loop form cycle steps progress statistics martin investigate model dynamics cellular functional state transition possible configuration ensemble cells edge connecting configuration configuration follows according rule properties structure number components length limiting cycles depth trees connecting non limiting states cycles instance incoming edge corresponds garden pattern self loop corresponds life pattern early application functional trains study triple systems train triple system functional having possible triple symbols triple mapped triples belong triple system contain pairs respectively trains shown powerful triple systems somewhat mathematical structure certain sets elements defined independent way independent sets satisfy properties modeled properties linear independence space standard examples graphic independent sets sets edges forests structure forests important computing minimum spanning tree define define edges set edges independent forms known bicycle smallest dependent sets minimal connected cycle called possible types bicycle connected internally paths figure consists cycles sharing single formed cycles connected path contain bicycle forbidden minors forming minor contracting edges produces family closed minors implies characterized terms finite set forbidden minors wagner having complete complete minors discussed non contains figure figure contracted form butterfly figure contracted form diamond non contains butterfly diamond minor minor minimal non butterfly diamond minor diamond butterfly resulting larger family consists unions multiple simply self loops considered forbidden minor loops early use involves network application flow problems modeling conversion different types problems given input flow network model edges model conversions edge marked capacity converted unit time flow conversion rate cost loss negative profit incurred unit conversion task determine convert edge flow network order minimize cost maximize profit capacity constraints allowing type accumulate unused type problem formulated linear program solved intermediate solutions arising eventual optimal solution special structure edge input network unused capacity edges forming spanning input network flow amounts lie zero capacity application called augmented trees called augmented forests minimum spanning problem involves finding spanning minimum weight larger edge weighted structure minimum weight found similar minimum spanning tree problem found efficient linear time approach case defined analogy minimum number edges minimum minimum edges oriented form directed structure time random edges chosen independently random possible pairs high probability constant strictly fact plays key role analysis data structure looking key value pairs looking tables locations determined key form correspond table locations edges link locations keys found succeeds finding locations keys play key role parallel coloring related problems
Pseudoforest