國立東華大學應用數學系國立東華大學應用數學系林 興 慶林 興 慶Lin-Shing-ChingLin-Shing-Ching指導教授指導教授 :: 郭大衛郭大衛

Vertex Ranking number of GVertex Ranking number of Graphsraphs

圖的點排序數圖的點排序數

A vertex ranking of a graph G is a mappinA vertex ranking of a graph G is a mapping g ff from from V(G)V(G) to the set of all natural num to the set of all natural number such that for any path between two dber such that for any path between two distinct vertices istinct vertices uu and and vv with with f(u)=f(v)f(u)=f(v) ther there is a vertex e is a vertex ww in the path in the path f(w)>f(u)f(w)>f(u).In this .In this definition,we call the value definition,we call the value f(v)f(v) the rank o the rank o

f the vertex f the vertex vv..

The vertex ranking problem is to deterThe vertex ranking problem is to determine the vertex ranking number mine the vertex ranking number r(G)r(G) of of

a given graph a given graph GG..

1 3 5 7 92 84 6

1 1 1 1 12 23 4

r(G)=4

Lemma If Lemma If HH is a subgraph of is a subgraph of GG,then ,then rr(H) ≦r(G)(H) ≦r(G) . .

The The unionunion of two disjoint graph of two disjoint graph GG and and HH is is the the G∪HG∪H

vertex set vertex set V(G∪H)=V(G)∪V(H)V(G∪H)=V(G)∪V(H) edge set edge set E(G∪H)=E(G)∪E(H)E(G∪H)=E(G)∪E(H)

Lemma Lemma 1

, ( ) max{ ( :1 )}k

i ii

If G G then G G i k

1 1 1 1 12 23 4

1 1 12 3

r(G1)=4

r(G2)=3

r(G1∪G2)=4

1 1 3 1 12 23 4

1

1

1

1

1

1

1

1

12

2 2

3

3 4

4 5

6

P9

P2××9

Lemma For any graph G,Lemma For any graph G, r(G)r(G)={min={minr*(Gs)r*(Gs):S is a minimal cut set of G}:S is a minimal cut set of G}

1 P12

2 23 4G˙S

1 1 1 1 1

32

P12

1 1 1

2 2 2 2 23 34 5G˙S

LemmaLemma For any graph For any graph GG,,r(G)r(G)==minmin{{r(G˙S)+1r(G˙S)+1:S :S isis an independent set of an independent set of GG}}

The The joinjoin of two graph G and H is the of two graph G and H is the GG++H H with with vertex set vertex set V(G+H)=V(G)∪V(H)V(G+H)=V(G)∪V(H)

edge set edge set E(G+H)=E(G) ∪E(H) ∪{xy:x in G,y in H}E(G+H)=E(G) ∪E(H) ∪{xy:x in G,y in H}

LLeemma mma ( ) min{ ( ) ( ) , ( ) ( )}r G H r G V H V G r H

r(G)=4

r(G+H)=r(G)+r(G+H)=r(G)+|V(H)|=6

1 1 3 1 12 23 4

P9

P2

GGrr is a graph withis a graph withV(GV(Grr)=V(G))=V(G)

E(GE(Grr))={={uvuv ｜｜ u,vu,v ∈∈V(G)V(G) and and ddGG(u,v)(u,v)≦r≦r}}

P72

1 12 2345

r(Pr(P7722)=5 )=5

PP303022

2

2log ( 1)

( ) log (1 ) 1

2

kn n

k

n n kr P k

k

TheoremTheorem For all For all n,kn,k with with n≧3n≧3 and and nn≧≧k-k-11

( ) ( )k kn n kC P k

Theorem For allTheorem For all

Theorem For allTheorem For all

21, ( ) log ( 1)nn P n

23, ( ) log 1nn C n

CCartesian product of two graph G and H is artesian product of two graph G and H is GG××HH V(G V(G××H)H)={(u,v)={(u,v) ｜｜ uu∈∈V(G)V(G) vv ∈∈V(H)V(H)}} E(GE(G××H)H)={(u,x)(v,y)={(u,x)(v,y) ｜｜ (u=v),xy(u=v),xy∈∈E(H)E(H) or(or(uvuv∈∈E(G),x=yE(G),x=y)})}

Try to Try to PP22××nn

P2××9

r(Pr(P22××99)=6 )=6

1

1

1

1 1

1

1

1

2

2 2

2 3

4

3

4

5

6

P2×2k+1×2k+1

P2×2k×2k

2 2 2

21 ( ) log 2 log

3n

nFor all n P P n

TheoremTheorem For a For a caterpillarcaterpillar T. Let T. Let PPnn be the be the subgraph of T obtained from T by subgraph of T obtained from T by deleting all leaves of T,and deleting all leaves of T,and { {vv ∈∈V(PV(Pnn):d):dGG≧3 ≧3 }={}={vvj1j1,v,vj2j2,…,v,…,vjkjk},}, where where jj11≦≦jj22≦… ≦j≦… ≦jkk.. IfIf we let we let jj00=0,j=0,jk+1k+1=n+1=n+1.. then then r(T)=r(Pr(T)=r(Pll)+1 )+1 wherewhere

11 1 1

1

1

2 2 2

ko k k i i

i

j j j j j jl k

1 1

1

1

1

1 1

1 1

1

1

v1

v2

v3

v4

v5

v6

1 1 1 1 1 11 1

2 2 2 5 23 42 23 3 34

( ) ( ) 1lr T r P

11 1 1

1

1

2 2 2

ko k k i i

i

j j j j j jl k

Composition of two graph,written Composition of two graph,written G[H]G[H]vertex set vertex set V(G) V(G) ×V(H)×V(H)

edge set edge set (u(u11,v,v11))is adjacent tois adjacent to(u(u22,v,v22))if eithher if eithher uu11 is adjacent to is adjacent to uu22 in G in G

G[H]=P4[P

5]

r(G[H])=r(Pr(G[H])=r(P44[P[P55])=r(P])=r(P44)+ 2|V(P)+ 2|V(P55)|=11)|=11

1

2

31

4

5

6

7

1

2

3

1

4

5

6

7

8

9

1011

Theorem For any two graphs G and Theorem For any two graphs G and H.H.

( [ ]) ( ) ( ) ( ( ) 1)G H H V H G

vertex setvertex set

edge setedge set

,11( , ; )

k

i i ii

H G v v

1

( ) ( )k

ii

V G V G

,1 ,11

( ) ( ) { ( )}k

i i j i ji

E G E G v v v v E H

V1,1 V2,1 V3,1

r(H1)=l r(H2)=l r(H3)=l

,11

( ( , ; )) 1n

i i ii

G H v v l

1 12 3

1 12 23

4

11

2

3

4

1 12 3

5

1 2

3

6

,1 11

( ( , ; )) ( ) 1n

i ii

G H v v G l ,1 1

1( ( , ; )) ( ) 1

n

i ii

G H v v G l

Corona of two graphs,writtenCorona of two graphs,written GGΛΛHH.. vertex set vertex set V(V(GΛHGΛH)=V(G) )=V(G) ∪{u∪{uijij|1|1≦i≦n,1≦j≦m≦i≦n,1≦j≦m}} edge set edge set

1

( ) { ( )}n

ij ik j ki

E G u u u u E H

11

{ 1 }n

iji

v u j m

21 1 21 1 21 1 21 1 21 1

1 12 23

Theorem For any two graphs G and Theorem For any two graphs G and H.H.

( ) ( ) ( )G H G H

3 34 45

感謝各位的參與感謝各位的參與