Rds To PCs I Rds I Hints - GitHub Pages
Transcript of Rds To PCs I Rds I Hints - GitHub Pages
level 2 introduction tograph theory
Notation if S is a set PCs thepomsetof S
setofsubsets ofS
S n PCs 2
Rds To PCs I T k
Rds I Hints
Red A simple graph is a pair GV E
where V is a nonemptyset and E is a subsetof PeCr
Unless we sayotherwise wewillassume that V is finite
Exe consider V graphs withvertexset 4 33
E Gas I G Ga differbyatmast on edge
I A
s I
Gwen agraph G CU E venikmile Uca fruE G fr E
Dd A subgraph Hofagraph G is a graph suchthat
VCH CUCA Ect CECH Nakhon Hc G
HCGDet A subgraphis is proje if
either Vlat VCH on
F G TECH
Det Asubgraph HCG is sparing if VCHVcd
If G is a graph Sc VCA we defy
GES subgraphw UCGCSI S
F GCDvertexinduced subgraph
ee Ela le c Bls
ECGapes
If X CE delle GEX subgraph induced by edgesX
E GER X V GID allwhus incidenttoedges in X
U eeEX
Def a subgraph HCG is called a component of G
if Hisconnected and if Heth c G H H
Hen H is notconnected
H is amaximal connected subgraph
Observation if H is a componentof G and ve UCH
let S we GI 7 awalk froma tow
then H GES
AHIG v G vCG lEu3
G e GEE G I e3 Csonetones sane
1 b 9but notalways
this sometimesremoves
thisnew vertices
removes a Igfvertices
G e V G e VIG
ECG e E G 163
Connectednesskcompouts
V 1,23 10
E 12 14 25 37 39 4,6107 32 56,59 62,71
412
i
Det Idol ofcomponents
A if G isconnected how can we quantify how
to
it is
y YETaptlyor
a nd
HE121 If G is
connected ee f Cod we say e is a bridge
if G eddisconnected
Manegenerally ifG is not
necessarily connected we say
ee EG is abridgeif it isbridge in oneofthecomps
IG
Alternately i e c Eco is abridgeif HackCG e
Det reVCG is a cutey ifHolck G D
Qi if e is abridge in aconnected graph G e Euw
then either G o or either u arw is abridge
Yesi if say u connectedto a f w
and G v isconnected then we hone a path
from u to w in G v but then wouldhave
a cycle a w Ev but if e is partofa cycle it can'tbe
a bridge
Therein if a vertex v is incident to a bridge e
then it is a cotvertex ifandonly if dey v z 2
Therein an edge ee Ecd is abridge ifandonly if
it is not on a cycle
theorem Ewy graphmusthave at least two entrees
which are not cot vertices
Det if u w e V G and P is a path from u to w
such that lengthofP is assmall as possible wesay
say P is a geodesicand we define d yw
length IP
theorem if G is aconnected graph no VCG u c UCG
whichis as far as possiblefromu then u is not a
cutartex