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Graphs khaled.nagi@alexu.edu.eg 2

hat is a Graph!

A graph G " # $%&' is co(posed of)$ ) set of *ertices&) set of edges connecting the *ertices in $

An edge e " #u%*' is a pair of *ertices&xa(ple)

$" +a%,%c%d%e-&"+#a%,'%#a%c'%#a%d'%#,%e'%#c%d'%#c%e'%#d%e'-

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Graphs khaled.nagi@alexu.edu.eg

&lectronic circuits

/ind the path of least resistance to 0S1

Applications # '

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Graphs khaled.nagi@alexu.edu.eg

3et4orks #roads% flights% co((unications'

Applications # '

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Graphs khaled.nagi@alexu.edu.eg 5

Scheduling #pro6ect planning'

(o7 ,etter exa(plesA Spike 8ee 9oint Production

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Graphs khaled.nagi@alexu.edu.eg

Ad6acent *ertices ) connected ,: an edgeDegree #of a *ertex ') ; of ad6acent *ertices

< deg#*' " 2#; edges'Since ad6acent *ertice each count the ad6oining edge% it 4ill ,e countedt4ice

Graph Ter(inolog: # '

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7/36Graphs khaled.nagi@alexu.edu.eg =

Path ) se>uence of *ertices *1%*2%. . .*k such that consecuti*e *ertices * iand * i?1 are ad6acent.

Graph Ter(inolog: # '

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8/36Graphs khaled.nagi@alexu.edu.eg

Si(ple path) no repeated *ertices

ore Graph Ter(inolog: # '

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9/36Graphs khaled.nagi@alexu.edu.eg B

0:cle) si(ple path% except that the last *ertex is the sa(e as the first*ertex

ore Graph Ter(inolog: # '

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10/36Graphs khaled.nagi@alexu.edu.eg 1C

0onnected graph ) an: t4o *ertices are connected ,: so(e path

Su,graph ) su,set of *ertices and edges for(ing a graph0onnected co(ponent ) (axi(al connected su,graph. &.g.% the graph,elo4 has connected co(ponents.

&*en ore Ter(inolog: # '

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11/36Graphs khaled.nagi@alexu.edu.eg 11

&*en ore Ter(inolog: # '

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12/36Graphs khaled.nagi@alexu.edu.eg 12

#free' tree connected graph 4ithout c:cles/orest collection of trees

E0ara(,aF Another Ter(inolog: SlideF

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13/36Graphs khaled.nagi@alexu.edu.eg 1

8et n " ;*ertices( " ;edges

0o(plete graph all pairs of *ertices are ad6acent( " #1 2'< deg#*' " #1 2'

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Graphs khaled.nagi@alexu.edu.eg 1

n " ;*ertices( " ;edges/or a tree ( " n 1

ore 0onnecti*it: # '

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Graphs khaled.nagi@alexu.edu.eg 15

If ( n 1% G is not connected

ore 0onnecti*it: # '

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Graphs khaled.nagi@alexu.edu.eg 1

A spanning tree of G is a su,graph 4hichIs a tree0ontains all *ertices of G

/ailure on an: edge disconnects s:ste( #least fault tolerant'

Spanning Tree

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Graphs khaled.nagi@alexu.edu.eg 1=

Ro,erto 4ants to call the TA7s to suggest an extension for the nextprogra(...

Jne fault 4ill disconnect part of graphFF A c:cle 4ould ,e (ore fault tolerant and onl: re>uires n edges

ATKT *s. RTKT#Ro,erto Ta(assia K Telephone'

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Graphs khaled.nagi@alexu.edu.eg 1

&uler and the Lridges of Moenigs,erg

0an one 4alk across each ,ridge exactl: once and return at the starting point!

0onsider if :ou 4ere a NPS dri*er% and :ou didn7t 4ant to retrace :our steps.In 1= % &uler pro*ed that this is not possi,le

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Graphs khaled.nagi@alexu.edu.eg 1B

Graph odel#4ith parallel edges'

&ulerian Tour ) path that tra*erses e*er: edge exactl: once and returns to thefirst *ertex&uler7s Theore( ) A graph has a &ulerian Tour if and onl: if all *ertices ha*e

e*en degreeDo :ou find such ideas interesting!ould :ou en6o: spending a 4hole se(ester doing such proofs!

Well, look into CS22!If :ou dare...

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Graphs khaled.nagi@alexu.edu.eg 2C

The Graph ADT is a positional container 4hose positions are the*ertices and the edges ofthe graph.

SiOe#' Return the nu(,er of *ertices plus the nu(,er of edges of G.Is&(pt:#'&le(ents#'Positions#'S4ap#'Replace&le(ent#'

3otation ) Graph G $ertices *% 4 &dge e J,6ect o3u($ertices#'

Return the nu(,er of *ertices of G.3u(&dges#'

Return the nu(,er of edges of G.

$ertices #' Return an enu(eration of the *ertices of G.&dges #' Return an enu(eration of the edges of G.

The Graph ADT

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Graphs khaled.nagi@alexu.edu.eg 21

Directed&dges#'Return an enu(eration of all directed edges in G.

Nndirected&dges#' Return an enu(eration of all undirected edges in G.

Incident&dges#*'

Return an enu(eration of all edges incident on *.InIncident&dges#*'

Return an enu(eration of all the inco(ing edges to *.

JutIncident&dges#*' Return an enu(eration of all the outgoing edges fro( *.

The Graph ADT #contd.' # '

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Graphs khaled.nagi@alexu.edu.eg 22

Degree#*'Return the degree of *.

InDegree#*'Return the in degree of *.

JutDegree#*'

Return the out degree of *.

The Graph ADT #contd.' # '

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Graphs khaled.nagi@alexu.edu.eg 2

Ad6acent$ertices#*'Return an enu(eration of the *ertices ad6acent to *.

InAd6acent$ertices#*'Return an enu(eration of the *ertices ad6acent to * along inco(ing edges.

JutAd6acent$ertices#*'

Return an enu(eration of the *ertices ad6acent to * along outgoing edges. AreAd6acent#*%4'

Return 4hether *ertices * and 4 are ad6acent.

&nd$ertices#e'Return an arra: of siOe 2 storing the end *ertices of e.

Jrigin#e'Return the end *ertex fro( 4hich e lea*es.

ore ethods ... # '

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Graphs khaled.nagi@alexu.edu.eg 2

Destination#e'Return the end *ertex at 4hich e arri*es.

IsDirected#e'Return true iff e is directed

ore ethods ... # '

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Graphs khaled.nagi@alexu.edu.eg 25

akeNndirected#e'Set e to ,e an undirected edge.

Re*erseDirection#e'S4itch the origin and destination *ertices of e.

SetDirection/ro(#e% *'

Sets the direction of e a4a: fro( *% one of its end *ertices.SetDirectionTo#e% *'

Sets the direction of e to4ard *% one of its end *ertices.

Insert&dge#*% 4% o'Insert and return an undirected edge ,et4een * and 4% storing o at this position.

Npdate ethods # '

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Graphs khaled.nagi@alexu.edu.eg 2

InsertDirected&dge#*% 4% o'Insert and return a directed edge ,et4een * and 4% storing o at this position.

Insert$ertex#o'Insert and return a ne4 #isolated' *ertex storing o at this position.

Re(o*e&dge#e'

Re(o*e edge e.

Npdate ethods # '

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Graphs khaled.nagi@alexu.edu.eg 2=

A GraphF Ho4 can 4e represent it!To start 4ith% 4e store the *ertices and the edges into t4o containers% and4e store 4ith each edge o,6ect references to its end*ertices

Additional structures can ,e used to perfor( efficientl: the (ethods of the

Graph ADT

Data Structures for Graphs

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Graphs khaled.nagi@alexu.edu.eg 2

The edge list structure si(pl: stores the *ertices and the edges intounsorted se>uences.&as: to i(ple(ent./inding the edges incident on a gi*en *ertex is inefficient since itre>uires exa(ining the entire edge se>uence

&dge 8ist

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Graphs khaled.nagi@alexu.edu.eg 2B

Operation Time

SiOe% is&(pt:% replace&le(ent% s4ap J#1'3u($ertices% nu(&dges J#1'

$ertices J#n'

&dges% directed&dges% undirected&dges J#('

&le(ents% positions J#n?('&nd$ertices% opposite% origin% destinationisDirected% degree% inDegree% outDegree

J#1'

Incident&dges% inIncident&dges% outIncident&dges%ad6acent$ertices% inAd6acent$ertices%outAd6acent$ertices% areAd6acent

J#('

Insert$ertex% insert&dge% insertDirected&dge% re(o*e&dge% (akeNndirected%re*erseDirection% setDirection/ro(% setDirectionTo

J#1'

Re(o*e$ertex J#('

Perfor(ance of the &dge 8ist Structure

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Graphs khaled.nagi@alexu.edu.eg C

Ad6acenc: list of a *ertex * )Se>uence of *ertices ad6acent to *

Represent the graph ,: the ad6acenc: lists of all the *ertices

Ad6acenc: 8ist #traditional'

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Graphs khaled.nagi@alexu.edu.eg 1

The ad6acenc: list structure extends the edge list structure ,: addingincidence containers to each *ertex.

The space re>uire(ent is J#n ? ('.

Ad6acenc: 8ist #(odern'

P f ( f h Ad6

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Graphs khaled.nagi@alexu.edu.eg 2

Operation Time

SiOe% is&(pt:% replace&le(ent% s4ap J#1'3u($ertices% nu(&dges J#1'

$ertices J#n'

&dges% directed&dges% undirected&dges J#('

&le(ents% positions J#n?('

&nd$ertices% opposite% origin% destination isDirected%degree% inDegree% outDegree

J#1'

Incident&dges#*'% inIncident&dges#*'%outIncident&dges#*'% ad6acent$ertices#*'% inAd6acent$ertices#*'% outAd6acent$ertices#*'

J#deg#*''

AreAd6acent#u% *' J#(in#deg#u'%deg#*'''

insert$ertex% insert&dge% insertDirected&dge% re(o*e&dge% (akeNndirected% re*erseDirection%

J#1'

Re(o*e$ertex#*' J#deg#*''

Perfor(ance of the Ad6acenc:8ist Structure

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Graphs khaled.nagi@alexu.edu.eg

Ad6acenc: atrix #traditional'

atrix 4ith entries for all pairs of *erticesQi%6 " true (eans that there is an edge #i%6' in the graph.Qi%6 " false (eans that there is no edge #i%6' in the graph.There is an entr: for e*er: possi,le edge% therefore)

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Graphs khaled.nagi@alexu.edu.eg

The ad6acenc: (atrix structures aug(ents the edge list structure 4ith a(atrix 4here each ro4 and colu(n corresponds to a *ertex.

Ad6acenc: atrix #(odern' # '

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Graphs khaled.nagi@alexu.edu.eg 5

Ad6acenc: atrix #(odern' # '

The space re>uire(ent is J#n 2 ? ('

Perfor( nce of the Ad6 cenc

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Operation Time

SiOe% is&(pt:% replace&le(ent% s4ap J#1'3u($ertices% nu(&dges J#1'

$ertices J#n'

&dges% directed&dges% undirected&dges J#('

&le(ents% positions J#n?('

&nd$ertices% opposite% origin% destination isDirected%degree% inDegree% outDegree

J#1'

Incident&dges% inIncident&dges% outIncident&dges%ad6acent$ertices% inAd6acent$ertices%outAd6acent$ertices%

J#n'

Are Ad6acent J#1'

Insert&dge% insertDirected&dge% re(o*e&dge%(akeNndirected% re*erseDirection% setDirection/ro(%setDirectionTo

J#1'

Insert$ertex% re(o*e$ertex J#n 2'

Perfor(ance of the Ad6acenc:atrix Structure