Graph Theory. Presentation Outline. Introduction. Introduction. Introduction

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1 Pesenttion Otline Gp Teo Apil 8 t, 4 Olie Kiki An Plo Intotion Fnmentl Conepts Repesenttions Speil Gps Bet-Fist Se Dept-Fist Se Topologil Se Stongl Connet Components Intotion Intotion Mn ee ptil polems n e pplie to n sole it gp teo. Clssil polems in gp teo e: Flo Connetiit (netok eliilit) Mting Elein lks (tesing eges one n onl one) Te Konigseg Bige Polem is often si to e te it of gp teo n gies goo si efinition of gp. In te it ie n tog te it s tt in its ente s n isln, n fte pssing te isln, te ie oke into to pts. Seen iges ee ilt so tt te people of te it ol get fom one pt to note. Te fone of netok topolog, Leon Ele, s inspie te qestion of ete it s possile to tese te entie it it onl ossing e ige etl one. Intotion Fnmentl Conepts In Ele s emple, e ln mss epesents ete of gp, onnete iges knon s eges.. Te egee of n ege is se on te nme of eges onneting to it. Te polem is no eomes one of ting to epesenttion of tis pite itot eting n lines. Ele onle fmosl in 76 in plise ppe tt tee s no soltion tis polem. Foml Definition: A gp G is tiple onsisting of ete set V(G), n ege set E(G), n eltion tt ssoites it ege to eties (not neessil sil istint) lle its enpoints. A ete is lle o if it s n o nme of eges leing to it, ote ise it is lle een. A simple gp is gp ing no loops o mltiple eges. Wen to eties e enpoints of n ege, te e jent A pt is simple gp ose eties n e oee so tt to eties e jent if n onl if te e onsetie in te list A le is gp it n eql nme of eties n eges ose eties n e ple on ile so tt to eties e jent if n onl if te ppe onsetiel long te ile. Simple Gp X is Ajent to X Pt Cle

2 Fnmentl Conepts Fnmentl Conepts A iete gp o igp G is gp in i te eges ontin infomtion petining te oeing o ietion of te ege fom m to jent eties. Te oee pi of eties ontin e n til. Te omplement G of simple gp G is te gp it sme ete set it te opposite onnetiit of eges A gp G is iptite if V(G) is te nion of to isjoint inepenent sets lle ptite sets of G A gp it no le is li. A foest is n li gp A tee is onnete li gp Foest Diete Gp G Complement of G Biptite people Tee jos Repesenttions Repesenttions Te jen mti of gp G is te n--n mti A(G) in i e ent [i,j] of te mti is te nme of eges in te gp it enpoints {[i{ [i], [j]} An jen-list epesenttion of gp G is n onsisting elements epesenting e ete of gp ontining list to fo ll te eties jent to it. Te eties in e list e tpill stoe in n it oe. Memo eqie is Θ(V + E) e W / / / / X Z W X Z Te sme onept of jen mties n list epesenttions n e pplie to igps f e g U V W X Z U / V W / X / / Z / Speil Gps Bet-Fist Se Te Petesen Gp is te simple gp ose eties e te - element ssets of 5-element 5 set n ose eges e te pis of isjoint -element ssets. Bsill it is speil gp tt s eties ll it egees of tee. One of te most si lgoitms fo seing gps n te iling lok fo mltifios ote gp lgoitms Sstemtill eploes te eges of gp in oe to isoe ee ete tt is ele fom ptil ete. Te en gol is to ompte te istne, o te lest nme of eges tt mst e tese, fom ete to e ele ete. Te nme omes fom te ft tt it epns te teito of knon n nknon eties nifoml oss te et of te fontie of te gp. E itetion of te se ils on te peios. Te lgoitm isoes ll eties t istne k fom te soe ete efoe isoeing n eties t istne k + Woks on ot iete n niete gps. A et-fist tee is ll ete t te sme tt ontins list of ll ele eties fom te soe.

3 Bet-Fist Se Bet-Fist Se Algoitm ses oloing seme to keep tk of pogess. All eties stt ot ite. Wen ne ete is isoee it goes to ge, on te net itetion it is lk. Te tee initill onl ontins te soe ete. Tee eges e igligte s te e poe te BFS. Te qee () is eset fo e itetion of te se s t s t s t s t s s t t s t t s t s t s t Ø Bet-Fist Se Dept Fist Se (DFS) Te opetions of en-qeing n e-qeing tkes O() Te jen list of e ete of te gp is snne onl en te ete s een e-qee, e is snne t most one. Sine te sm of te lengts of ll jen lists Θ(E) te totl time spent in snning jen lists is O(E) Te oee fo initilition is O(V) An te totl nning time of BFS is O(V + E) Bsi Ie: Selet stting ete n eploe s f s possile long e n efoe ktking to eploe te est of te noes in te gp. If tee e n eties left tt e nele fom te stting ete, one of tem is osen s te ne stting point n te DFS lgoitm is pplie to tis pt of te gp. DFS lgoitm ses timestmps to mk e ete, ts keeping tk of te time en te ete s fist isoee ([]) n en it s isite gin s pt of te ktking poess (f[]). Timestmps e integes eteen n V, ee V is te nme of eties. [] < f[] Θ (V+E) Te DFS Algoitm DFS (G) fo e ete V[G] o olo[] <-< WHITE //fist ll noes e ite π [] <-< NIL //none e een isoee et //time ts s onte fo e ete V[G] o if olo[] = WHITE ten DFS-VISIT () Te DFS Algoitm () DFS-VISIT () olo[] <-< GRA time + [] <-< time fo e Aj [] o if olo[] = WHITE //nge noe olo //isoe timestmp //eploe ne ege ten π [] <-< //mk te pt DFS-VISIT () //esion olo[] <-< BLACK //If is te en, mk it lk f[] <-< time+ //enote s // ktking timestmp

4 DFS Popeties DFS Gp Emple DFS foms foest of tees, i n e eonstte oseing esie lls to te DFS-VISIT () fntion n te oloing of eties tt it poes. Te tees e ilt sing timestmp les in oe to fige ot te esenents of noes Vete is pope esenent of ete if [] < [] < f[] < f[] In ote os, is isoee efoe n mke fte ing ktking /6 4/5 s t /9 / /6 7/8 / 4/5 s t (s ( ( ( ) ) ( ) ) s) (t ( ) ( ) t) Clssifition of Eges DFS n e se to lssif te eges of te gp Fo possile tpes: Tee eges: (,) is tee ege if s fist isoee eploing ege (,) Pts of ept fist foest Bk eges: (, ) is k ege if it onnets ete to n nesto Self-loops loops Fo eges: (, ) is fo ege if it onnets ete to esenent in ept-fist tee Coss eges: All ote eges. Cn go eteen eties in te sme tee, o to eties in ote ept-fist tees. Topologil Sot Cn onl e pplie fo iete li gps (DAG) Tkes DAG, n etns line oeing of ll its eties, s tt fo n ege (, ) ete ppes efoe in te oeing DAG s m inite peeene mong some eents i e not neessil elte, ts topologil sot n e se to onstt n oell oe of eents. TOPOLOGICAL SORT (G) DFS (G) poes timestmps f[] fo ll eties in gp s e ete eeies its timestmp, it to linke list Rns in time Θ (V+E) DFS is Θ (V+E) Linke List opetions e O () Topologil Sot Stongl Connete Components /6 eet tong soks 7/8 t 9/ /5 pnts soes /4 sit /8 6/7 elt tie /5 jket /4 soks tong pnts soes t sit elt tie jket Using DFS to eompose iete gp into its stongl onnete omponents Stongl onnete omponents e sets of eties C itin gp G, s tt te e ete is essile fom ete n ie es, ee, C. (Loops) In oe to fin te stongl onnete omponents of gp, e fist mst ompte its tnspose. Tnspose of iete gp G n e togt of s iete gp Gt,, i ontins ll te sme eties V s G, et te ietions of ll eges E G e eese. G n Gt e etl te sme stongl onnete omponents. Te ompttion of stongl onnete omponents is pefome in line time Θ (V+E) nning to DFS, one on te gp G n one on its tnspose Gt. 4

5 Stongl Connete Components Stongl Connete Components STRONGL CONNECTED COMPONENTS (G) /4 /6 /5 /4 e f e fg / /7 g 8/9 5/6. DFS(G) //ill ompte timestmps f[] fo e //ete in gp G. Compte (Gt)( //Tis tkes Line time. DFS(Gt) //Consie te eties in oe of //eesing f[], s ompte in step 4. Otpt te eties of e tee in te ept-fist foest fome in step. //Lst step ill ispl te stongl //onnete omponents in gp G Refeenes West, Dogls B. Intotion to Gp Teo, Pentie-Hll, In., Rosen, Kennet H. Disete Mtemtis n Its Applitions, MG-Hill, 999 5