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# Graph - Part I (Logical Level) CSS 343

UW

GPA 3.5

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## Popular in Data Structures, Algorithms, and Discrete Mathematics II

## Popular in ComputerScienence

This 9 page Study Guide was uploaded by NGHIEP NGO on Thursday October 8, 2015. The Study Guide belongs to CSS 343 at University of Washington taught by Dr. Min Chen in Fall 2015. Since its upload, it has received 161 views. For similar materials see Data Structures, Algorithms, and Discrete Mathematics II in ComputerScienence at University of Washington.

## Popular in ComputerScienence

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Date Created: 10/08/15

Grapha Legiel Leeel Binary treea prpaeitle a very useful erar pl representing relatipnahipa in whieh a hierar ehy reaziata That la a nude is painted ta hr at meet ape lather nestle 1 s parentj and eaeh Itprle paints the at meat twu ether harlea ite Children If we remave the reatrietipn that eaeh IIIEIIElE ean have at meat twla Ell ihdf we have a general tree as pictured here If we also remove the restriction that each node may have only one parent node we have a data structure called a graph A graph is made up of a set of nodes called vertices and a set of lines called edges or arcs that connect the nodes IEraph h data atrhetape that eheiate at a set It Vapaea and a pet tertpea that relate the ripdea tp rf E anather Vertex It tle in a graph Edge are H paur at FEPTEEEEMlF Ig a e pep eetiph het heee twp hpdea in a graph Uf r lr t d graph A graph irn whireh the eagea have rue drreetlph l ireetetl graph igraphl A graph nr erhieh eaeh leee i5 areeteel harp prie pertea tp apather ler the aarnei vertex Undirected Graph The set of edges describes relationships among the vertices For instance if the vertices are the names of cities the edges that link the vertices could represent roads between pairs of cities Became the road that mnr hetneen Honrton andAnrtz39n 6150 mnr hetneen Anrtzn and Honrton the edger in the graph hane no direction This structure is called an undirected graph a Graph l i5 an unt reeted graph teerape a H e h El ephli re 1 a e r e Is He Directed Graph However if the edges that link the vertices represent ights from one city to another the direction of each edge is important The existence of a ight edge from Houston to Austin does not assure the existence of a ight from Austin to Houston A graph Whose edges are directed from one vertex to another is called a directed graph or digraph lb Graph is a dieEmit helpli 41 ample i5 3 ire mph nial Ir IE MILEr E E H I 1 EIEFHIIIIWEII HE D E I EDI E III H I H L 11 W A H111 EFL EH 131th 13 5 Le M EI39E39III39EIIIIIHJI tn 33L 3 5 quot 5 53quot 9 311193 11 1 Figure 93 Same exempt fymphs me h a prguremmeur39e perepeetive u em39tieee repreeermt whatever ie the Ehhjeet f Uh Shady pexuapele heaueee ei ljee retzr lhleheer and Et an Mhthematieelly eertie are the untie he E it lE p t uptm whieh graph thenmfg rate F i i l j a graph G i5 defined he analluwe E E WEquot h a nite nunemipir Set fvertfeee HIE j 5 Settf as pain ethertieesj T specify the f verljeee list them ih eet hetatin within hmem The fealluwingi eel tilefinm the ner if the graph piEMI i in ELIE i litGraph W Hr E 3 J The suit edge5 i5 epeei eti by listing a eeq hehee hf edge5 Tee dermete eaeh edge write the HEIDIE5 suit the tw it enhneets in paten thmee with e eemma between them Fur ineteeee the in Gmphl in Figure 9171 are enihneetetl hy the lth edg described helew HEM I M h m h h h h h he 2 T J L Q 11 315 e deem HEM I e e BLAH11 at TD T EU 1 i 1 Adjacent vertices If Z2920 ee z39eee in a grep9 are eomeeeled by me edge they are said to be adjacent In Graphl Figure 98a vertices A and B are adjacent but vertices A and C are not a Emph1 i5 en undireeted graph weemeI e H e U empire he L Mr I r e m was If the vertices are eomeeeled by a directed edge then the rst vertex is said to be adjacent to the second and the second vertex is said to be adjacent from the first For example in GraphZ in Figure 98b vertex 5 is adjacent to vertices 7 and 9 While vertex 1 is adjacent from vertices 3 and 11 h Graph is a directed graph W mph 1 33 E 1 911 E ir ph H1 3 3 1II E if 5 9 11r LEE 9 11 1E Adjeeeht vertices Thel eer tiees in a graph that are een HEC EEEI quota an edge Path A Eerghenee hf verheee that E EETE the melee ih a graph Cempliei e grph A graph ih which eraserF1 Herrera i5 dureetl e ehhheeted h were ether eerteh Weighted graph 5 graph 391 whieh earsen edge earriee 3 value Path Complete Graph Weighted Graph A path fmrn une vertex te anather euguaieta eat a elf EFEl39 EE that ennheet them Far a path ta exiat an uninterrupted eequenee at edge5 must gr hen the rat vertex threug h ef verlieea tn the eeeunrl Fer exampte in Graph a path ghee tram vertex 5 tr 3quot but nt fmgrrl vertex 3 tr vertex 5 Hate that in a tree aueh as Graph Figure ne a unique path exiate them the met te EquotIurerjir ether nude in the In a vertex ia adjaeeht tu eateryr ether vertex Figure 93 shewe twe eamp ete graphs If 1here are N vertieea there wit N 39 N M in a emphete heeterl graph and N quot N I I E edges in a eufrrlplete undirected graph In a eaeh edge earriea a value Weighted rapha ean he need t rep resent apphieatiuua in w ieh Ihe i f ef the euuneetinn between the vertieea is impr tant neatjnet the at a euuneetinn Far inatanee in the graph pietu in Fiure 1111 the vertiem repireaeut iti and the quotan indicate the Air Euetere Air n lines ighm that eenueet the ei ea The weighta attaehed tr the edgea repaireaeut the air j t t between pairs F ei ea a Eamplete dim graph h famplete umdireeterl graph Figure 99 Twa eaman etey mphs Harlem Figure He llE A WEme graph Eraph AEIT 1 i ti i EtImEture The graph E f l i hf a 5E1 Elf 39HEHiEEE and 3 SEE if weightEd ErEEE that E 39l lEE E 5mm Dar all if 1111 waning m 3an anther perah n A ump n BEEFEsra any 31 i5 I d 1 a graph rp39EITEI DIL 39IIhE graph hag been declared and a Ebi atmtt f hag h E EI l app i du MEHEEITIEIIH F c m II IiIiEIjEE HE gra p39l m an Empty 5mm P m iadf n Graph i5 emptjg men Empltjpr F f TrEEtE whE EhEr IE graph i5 empty P r mdf Functinn yaqu graph i5 Empty Bataan HEPLIM Fum ru Team WhE IE thE graph is u a Pastmmii a Fu m iiun value graph i5 mm MUEHEEIWEHEKTWE werltexj Fum im Adda vert x m the graph Graph i5 Tit l Pusim di n is in VEraph MEdEJWEHEETWE frm iui erltmt Vertemwpe t D39VEF II39EJII Edge iui a uequpe W Eig h ltjl Fum ru Adda an Edge with 39lE Speci d w g h Emm Emm VErtex tn t VErtex EmmVErtEx aan this me are in ngraph Pustmm l i fmmZVEJTtEL WHERE i5 in Eigmph with 12h Epfti E weight EdQEM lUETE E WEJQHEIEWMI EEETHPE f rmvert x UEF II39EITyfpi j lt iurert x Fum ru EtE I Jj EE HIE weight snaf39 the gig Emm Emm VErtex tn ttv rtex Emm VErtex and 1me are in ngpah Pustmmii s Fll ii yamE WEight If Emm Emm VErtEx I IisuaVEriiEx if Edge If E g d ht EKiEL msztinn val pEEi l quotm l Edg vahu Et TVEaFtiEEEWE IEKTSWE var lz39ex ueTypeEa werltex j Fum rm Rama 3 LIELlE if ue HEHIiEEE that are adja EEII II fmm vertex i5 in VEraph Pmtmmii u tuning the 1131111555 hf 31h vErtiEEE that ME dj t mm Hermit Connected amp Unconnected Graph Connected Graph 39 A connected graph has a path from every vertex to every other vertex 39 To recognize this kind of map notice that all nodes need to have both in and out ow 8 12 has no out then this is an unconnected graph Biconnected amp Non biconnected Graph 39 A connected undirected graph Where no vertices Whose removal disconnects the rest of graph I Applications Network mail mass transit system This is bi connected graph Hunhi E inmateE 1 Ed Tree When remove C graph become disconnected Articulation Point 39 Root is articulation if it has more than 1 child 39 Other node V is articulation if its one of the child W has 10WWgtseqv Example 1 v hf if f lf h DEE Swimming TIEE 11 1 E tnnn tte Tree Figf u a If quot0 III quothe I III 39iu 339 11 Hi 39Ia If I I ll f r 39i i39 I i I 39 I J 39 I I i l gt u u 1 i 3 11 I5 11 l V Eeq EEEIIMJEWEE I i by taking Ell m mm Edgaa amid iihiamm E it y me back edge Example 2 In h FiIEt E lr h DEE Swimming TEEE 1 1 thrhd unn te Tie Example 3 DEmth Fir t Eearch DEE Swimming TIEE V 111 W 1quot thrhi nnn te Ike

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