Rds To PCs I Rds I Hints - GitHub Pages

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level 2 introduction to graph theory Notation if S is a set PCs the pom setof S set of subsets of S S n PCs 2 Rds To PCs I T k Rds I Hints Red A simple graph is a pair G V E where V is a nonempty set and E is a subset of Pe Cr Unless we say otherwise we will assume that V is finite Exe consider V graphs with vertex set 4 33 E Gas I G Ga differ by at mast on edge I A s I

Transcript of Rds To PCs I Rds I Hints - GitHub Pages

Page 1: 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

Page 2: Rds To PCs I Rds I Hints - GitHub Pages

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

Page 3: Rds To PCs I Rds I Hints - GitHub Pages

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

Page 4: Rds To PCs I Rds I Hints - GitHub Pages

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

Page 5: Rds To PCs I Rds I Hints - GitHub Pages

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

Page 6: Rds To PCs I Rds I Hints - GitHub Pages

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