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Excursions in Modern Mathematics, 7e: 5.6 - 2Copyright © 200 !earson Education, "nc.

5 The Mathematics of GettingAround

5. Eu#er Circuit !ro$#ems

5.2 %hat "s a &raph'

5.( &raph Concepts and )ermino#ogy

5.* &raph Mode#s

5.5 Eu#er+s )heorems

5.6 Fleury’s Algorithm

5.7 Eu#eriing &raphs


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%hen the graph has an Eu#er

circuit orpath, ho do e ind it' /or sma## graphs,

simp#e tria#-and-error usua##y ors ine, $ut

rea#-#ie app#ications sometimes in1o#1e

graphs ith hundreds, or

e1en thousands,

o 1ertices. "n these cases a tria#-and-error

approach is out o the uestion, and hat is

needed is a systematic strategy that te##s usho to create an Eu#er circuit or path. "n

other ords, e need an a#gorithm.

Algorithms


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)here are many types o pro$#ems that can$e so#1ed $y simp#y o##oing a set o procedura# ru#es31ery speciic ru#es #iewhen you get to this point, do this, ... after

you finish this, do that , and so on. &i1en aspeciic pro$#em X , an algorithm or so#1ing

X is a set o procedural rules that, heno##oed, a#ays #ead to some sort o4so#ution to X .

X need not $e amathematics pro$#em3a#gorithms are used,sometimes unitting#y, in a## a#s o #ie:

Algorithms


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directions to ind someone+s

house, theinstructions or assem$#ing a ne $ie, or arecipe or $aing an app#e pie are a##examp#es o rea#-#ie a#gorithms. useu#

ana#ogy is to thin o the

pro$#em as a dishe ant to prepare and the a#gorithm as arecipe or preparing

that dish.

"nmathematics, a#gorithms are either ormu#a

dri1en you 8ust app#y the ormu#a orormu#as to the

appropriate inputs9 ordirecti1e dri1en you must o##o a speciicset o directi1es9.

Algorithms


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%e i## no turn our attention to ana#gorithm that inds an Euler circuit or an Euler path in a connected graph.)echnica##y speaing, these are to

separate a#gorithms, $ut in essence theyare identica#, so they can $e descri$ed asone.

)he idea $ehind /#eury+s a#gorithm can $eparaphrased $y that o#d piece

o o#isdom: Don’t burn your bridges behindyou.

Fleury’s Algorithm


7/30Excursions in Modern Mathematics, 7e: 5.6 - 7Copyright © 200 !earson Education, "nc.

"n graph theory the

ord $ridge has a 1ery speciic

meaning3it is the on#y edge

connecting to

separate sections

ca## them


A and B9

o a

graph, as

i##ustrated

in /ig.5-.



)hus, /#eury+s a#gorithm is $ased on asimp#e princip#e: )o ind an Eu#er circuit or

an Eu#er path, bridges are the last edges

you want to cross. ;ur concerns #ie on#y on

ho e are going to get

around the yet-to-

be-traveled part o the graph. )hus, hen

e ta# a$out $ridges that e ant to #ea1e

as a #ast resort,e are rea##y reerring tobridges of the to-be-traveled part of the

graph.




< Preliminaries. Mae sure that the

graph is connected and either 9 has

no odd 1ertices circuit9 or 29 has 8ust

to odd 1ertices path9.

< Start. Choose a starting 1ertex. ="n

case 9 this can $e any 1ertex> in

case

29 it must $e one o the to odd1ertices.?

/@EAB+D @&;B")M /;B

/"FG"F& F EA@EB C"BCA") !)9



< Intermediate steps. t each step, iyou ha1e a choice, don+t choose a

$ridge o the yet-to-$e-tra1e#ed part

o the graph. oe1er, i you ha1eon#y one choice, tae it.

< End. %hen you can+t tra1e# any

more, the circuit path9 is comp#ete.="n case 9 you i## $e $ac at the

starting 1ertex> in case 29 you i##

end at the other odd 1ertex.?

/@EAB+D @&;B")M /;B

/"FG"F& F EA@EB C"BCA") !)9



"n imp#ementing /#eury+s a#gorithm it is

critica# to separate

the past

the part o the

graph that has a#ready $een tra1e#ed9 rom

the uture the part o the graph that sti##

needs to $e tra1e#ed9. %hi#e there are manydierent ays to accomp#ish this you are

certain#y encouraged to come up ith one

o

your on9, a air#y re#ia$#e ay goes #iethis: Dtart ith to copies o the graph.

Copy is to eep trac o the 4uture> copy

2 is to eep trac o the 4past.

Fleury’s Algorithm Bookkeeping



E1ery time you tra1e# a#ong an edge, erase

the edge rom copy , $ut mar it say

in

red9 and #a$e# it ith the appropriate

num$er on copy 2. s you mo1e orard,

copy gets sma##er and copy 2 gets redder. t the end, copy has disappeared> copy 2

shos the actua# Eu#er circuit or path.

Fleury’s Algorithm Bookkeeping



)he graph in /ig. 5-Ha9 is a 1ery simp#e

graph 3 it ou#d $e easier to ind an Eu#er

circuit 8ust $y tria#-and-error than $y using

/#eury+s a#gorithm. Fonethe#ess, e i## do it

using /#eury+s a#gorithm. )he rea# purpose o

Examp#e 5.20 "mp#ementing /#eury+s

#gorithm

the examp#e is to see the

a#gorithm at or.Each stepo the a#gorithm is exp#ained

in /igs. 5-H$9 through h9.



Start: %e can pic any starting point e

ant. @et+s say e start at F .


#gorithm


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Step 1: )ra1e# rom F to . Cou#d ha1e a#so

gone rom F to D.9


#gorithm


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Step : )ra1e# rom to D. Cou#d ha1e a#so

gone to A or to E .9


#gorithm


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Step !: )ra1e# rom D to A. Cou#d ha1e a#so

gone to B $ut not to F ! DF is a $ridgeI9


#gorithm


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Step ": )ra1e# rom A to . Cou#d ha1e a#so

gone to E $ut not to B 3 AB is a $ridgeI9


#gorithm


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Step 5: )ra1e# rom to E . )here is no

choiceI9


#gorithm


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Steps 6# $# %# and &: ;n#y one ay to goat each step.


#gorithm


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Dince it ou#d $e a #itt#e impractica# to sho

each step o the a#gorithm ith a separate

picture as e did in Examp#e 5.20, you are

going to ha1e to do some o the or. Dtart$y maing to copies o the graph.

Examp#e 5.2 /#eury+s #gorithm or

Eu#er !aths

< Start. )his graph has to odd 1ertices, E and " . %e can pic either one as the

starting 1ertex. @et+s start at " .


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< Step 1. /rom " e ha1e i1e choices, a## o

hich are ;J. %e+## random#y pic # .

Erase "# on copy , and mar and #a$e#

"# ith a on copy 2.9


Eu#er !aths

< Step 2. /rom # e ha1e three choices B, $, or % 9. ny o these choices is

;J. Day

e choose B. Fo erase #B rom copy

and mar and #a$e# #B ith a 2 on copy 2.9


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< Step 3.

/rom B e ha1e three choices A,

, or " 9. ny o these choices is ;J. Day

e choose . Fo erase B rom copy

and mar and #a$e# B ith a

( on copy 2.9


Eu#er !aths

< Step 4. /rom e ha1e three choices D, E , or $9. ny o these choices is

;J. Day

e choose $. EM@3that+s shorthand or

erase, mar, and #a$e#.9


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< Step 5.

/rom @ e ha1e three choices E ,

&, or # 9. ny o these choices is ;J. Day

e choose # . [email protected]


Eu#er !aths

< Step 6. /rom # e ha1e on#y one choice3to . %e

choose % . [email protected]


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< Step 7.

/rom e ha1e three choices &,

' , or " 9. %e


Eu#er !aths

shou#d not

choose &, as

%& is a $ridge

o the yet-to-$e-

tra1e#ed part othe graph see

/ig. 5-29.


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< Step 7.

Either o the other to

choices is

;J. Day e


Eu#er !aths

choose K.

[email protected]


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< Step 8.

/rom " e ha1e three choices A,B, or ' 9, $ut e shou#d not choose ' , as "' has 8ust $ecome a $ridge. Either o the

other to choices is ;J. Day e

choose B.EM@9


Eu#er !aths

< Step 9 through 13. Each time e ha1eon#y one choice.

/rom B e ha1e to go to

A, then to " , ' , % , and &.


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< Step 14 through 21.

Fot to $e#a$or thepoint, #et+s 8ust cut to the chase. )he rest othe path is gi1en $y &, F , E , D, , E , &, $,

E . )here are many possi$#e endings, and you shou#d ind a dierent one $y yourse#.


Eu#er !aths

)he comp#eted Eu#er path one o hundreds

o possi$#e

ones9 is shon in /ig. 5-22.


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Eu#er !aths

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