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On On cc-Vertex Ranking of Graphs -Vertex Ranking of Graphs

Yung-Ling Lai & Yi-Ming ChenNational Chiayi UniversityTaiwan

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Vertex Ranking Vertex Ranking f f : : VV((GG)){1,2, …,{1,2, …,kk}}

ff ( (vv) = ) = ff ( (uu) ) Every Every uu--vv path, path, ww such that such that ff ( (ww) > ) > ff ( (vv) )

kk--rankablerankable vertexvertex ranking number ranking number rr((GG))

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CC88 is is 55--rankablerankable;; rr((CC88)=4)=4

1 2

2 1

4

1

1

3

1 2

4 1

5

3

3

2

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cc-vertex ranking-vertex ranking

vertex rankingvertex ranking every connected component has at most one vertex every connected component has at most one vertex

with maximum labelwith maximum label

cc-vertex ranking-vertex ranking every connected component has at most every connected component has at most cc vertices vertices

with maximum label with maximum label

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c-Vertexc-Vertex RankingRanking f f : : VV((GG)){1,2, …,{1,2, …,kk}}

After removing the vertices with maximum rAfter removing the vertices with maximum rank, each component of the remaining graph ank, each component of the remaining graph has no more than has no more than cc vertices with maximum r vertices with maximum rank ank

kk-c--c-rankablerankable c-vertexc-vertex ranking number ranking number rrcc((GG))

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Example (Example (cc-vertex ranking)-vertex ranking)

1 3 1 2 1 4 1 2 3 1

1 1 3 1 2 3 1 2 2 1

vertex ranking

c-vertex ranking with c = 2

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TheoremTheorem

The c-vertex ranking number of path The c-vertex ranking number of path PPnn is is

1( ) log ( 1)c n cr P n

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PathPath

1 1 1 12 21 1 11 23

c c c c

1max rank log ( 1)cx n

Rank only increase when Rank only increase when ii is the power of is the power of cc+1+1 When rank increase to When rank increase to x x n n ( (cc+1)+1)xx-1-1

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Analysis - Path Analysis - Path Number of VerticesNumber of Vertices Maximum RankMaximum Rank

1 ~ 1 ~ cc 11

cc+1 ~ +1 ~ cc++cc((cc+1)+1) 22

cc++cc((cc+1)+1~ +1)+1~ cc+[+[cc++cc((cc+1)](+1)](cc+1)+1) 33

==cc++cc((cc+1)++1)+cc((cc+1)+1)22

…………The minimum of the maximum rank The minimum of the maximum rank xx has value has value

1

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(( 1) 1)( 1) log ( 1)

( 1) 1

xxi

ci

c cc c n x n

c

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Theorem Theorem

The The cc-vertex ranking number of cycle -vertex ranking number of cycle CCnn is is

1( ) log 1nc n c cr C

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Cycle Cycle

1 1 1 12 21 1 113

c c c c

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If the rank of If the rank of vvnn (say (say xx) don’t need to increased, th) don’t need to increased, th

ere are no more than ere are no more than cc vertices with rank vertices with rank xx in in PPnn nn ( (cc+1)+1)xx-1-1

1( 1)1 1log 1 log 1

xcnc cc c

2

1log ( 1) 1 1xc c x

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Cycle Cycle

1 1 1 12 21 1 113

c c c c

1log 1nc c

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If the rank of If the rank of vvnn has to be increased (to has to be increased (to xx), ther), ther

e are e are cc vertices with rank vertices with rank xx-1 in -1 in PPnn-1-1 nn-1 -1 cc((cc+1)+1)xx-2-2

2( 1) 11max rank log 1

xc cc cx

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Analysis - CycleAnalysis - Cycle No more than No more than cc vertices with maximum rank vertices with maximum rank Remove those vertices with maximum rank will result Remove those vertices with maximum rank will result

no more than no more than cc paths paths At least one of the path must have no less than (At least one of the path must have no less than (nn--cc)/)/cc

verticesvertices

that path needs at least that path needs at least

ranksranks

The whole cycle needs at least The whole cycle needs at least ranksranks

1log ( 1)n cc c

1log nc c

1log 1nc c

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TheoremTheorem

The c-vertex ranking number of wheel The c-vertex ranking number of wheel WWnn is is

1 1( ) log 1nc n c cr W

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Wheel (Wheel (WWnn==KK11++CCnn))

1 1

1

1

1

1

2

2

3

3

If rank of If rank of vvcc (say (say xx) is t) is t

he same as max rank in he same as max rank in path path PPnn-1-1 then then

1( 1)xn c

1

1 1

( 1)1 1

log 1

log 1x

nc c

cc c x

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Wheel (Wheel (WWnn==KK11++CCnn))

1 1

1

1

1

1

2

2

3

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If rank of If rank of vvcc (say (say xx) is one ) is one

more than max rank in patmore than max rank in path h PPnn-1-1 then there are at leasthen there are at leas

t t cc-1 vertices ranked as -1 vertices ranked as xx--1 in the path 1 in the path

21 ( 1)( 1)xn c c

2

1 1

( 1)( 1) 11 1

log 1

log 1x

nc c

c cc c x

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Analysis - Wheel Analysis - Wheel

Case 1: Case 1: vvcc is the only vertex with max rank is the only vertex with max rank

Since Since 1( ) log 1nc n c cr C

1 1 1( ) log 2 log 1n nc n c cc cr W

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Analysis - Wheel Analysis - Wheel Case 2: Case 2: vvcc is not the only vertex with max rank is not the only vertex with max rank The graph after removing the vertices with maThe graph after removing the vertices with ma

x rank is a collection (no more than x rank is a collection (no more than cc-1) of pat-1) of pathshs

There is a path with at least verticesThere is a path with at least vertices

( 1)1

n cc

( 1)1 11 1( ) log 1 logn c n

c x c cc cr P

1 1( ) log 1n

c n c cr W

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Analysis - Wheel Analysis - Wheel Case 3: Case 3: vvcc is not ranked with max rank (say is not ranked with max rank (say xx)) Remove the vertices with max rank won’t separRemove the vertices with max rank won’t separ

ate the graphate the graph Assume remove vertices with rank greater than Assume remove vertices with rank greater than

xx--y y will separate the graph to no more thanwill separate the graph to no more than cy cy-1 -1 pathspaths

There exists a path containing at leastThere exists a path containing at leastverticesvertices

( 1)1

n cycy

( 1)1 11 1( ) log 1 logn cy n

c x c ccy cyr P

1 11 1( ) log log 1n nc n c ccy cr W y

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TheoremTheorem

The The cc-vertex ranking number of complete bipa-vertex ranking number of complete bipartite graph rtite graph KKm,nm,n for for mm nn is is

,

1 ;( )

1 .c m n mc

if m n cr K

otherwise

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Complete Bipartite GraphComplete Bipartite Graph

all vertices call vertices can be ranked as 1an be ranked as 1

All vertices iAll vertices in n mm partite set ranked as 1 partite set ranked as 1

The vertices in The vertices in nn partite set partite set

ranked as 2 to ranked as 2 to

m n c

m n c n

m

2n

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Corollary Corollary

The The cc-vertex ranking number of complete -vertex ranking number of complete rr-pa-partite graph for is rtite graph for is

1 2, ,..., rn n nK 1 2 .... rn n n

1

1 1( ) min 1,r r

i ii ic

n nr Gc c

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Thank youThank you