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1 On c-Vertex Ranking of Graphs Yung-Ling Lai & Yi-Ming Chen National Chiayi University Taiwan

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2 Vertex Ranking f : V(G) {1,2, …,k} f (v) = f (u) Every u-v path, w such that f (w) > f (v) k-rankable vertex ranking number r (G)

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3 C 8 is 5-rankable; r (C 8 )=4 12 21 4 1 1 3 12 41 5 3 3 2

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4 Background Studied since 1980s For an arbitrary graph G is NP-complete Two versions of vertex ranking problem off-line on-line

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5 On-line Vertices are given one by one in an arbitrary order Know the edges of the induced subgraph Assigned vertex ranking in real time Cannot be changed later Used denoted on-line vertex ranking number

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6 Example ( On-line) 1 1 1 11 2 2 3 T Off-line 3 2 2 34 1 1 5 T On-line v1v1 v2v2 v7v7 v4v4 v8v8 v3v3 v5v5 v6v6 n=8

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7 Bounds in online version

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8 c-vertex ranking vertex ranking every connected component has at most one vertex with maximum label c-vertex ranking every connected component has at most c vertices with maximum label

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9 c-Vertex Ranking f : V(G) {1,2, …,k} After removing the vertices with maximum rank, each component of the remaining graph has no more than c vertices with maximum rank k-c-rankable c-vertex ranking number r c (G)

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10 Example (c-vertex ranking) 1312141231 1131231221 vertex ranking c-vertex ranking with c = 2

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11 Application VLSI layout problem designing communication network planning efficient assembly of products in manufacturing systems Minimum height of the separator tree

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12 Example (separator tree ) 1 1 1 11 2 2 3 T

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13 Theorem The c-vertex ranking number of path P n is

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14 Path 111122111123 cccc Rank only increase when i is the power of c+1 When rank increase to x n (c+1) x-1

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15 Analysis - Path Number of VerticesMaximum Rank 1 ~ c1 c+1 ~ c+c(c+1) 2 c+c(c+1)+1~ c+[c+c(c+1)](c+1)3 =c+c(c+1)+c(c+1) 2 …… The minimum of the maximum rank x has value

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16 Theorem The c-vertex ranking number of cycle C n is

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17 Cycle 11112211113 cccc 3 If the rank of v n (say x) don ’ t need to increased, there are no more than c vertices with rank x in P n n (c+1) x-1

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18 Cycle 11112211113 cccc 3 If the rank of v n has to be increased (to x), there are c vertices with rank x-1 in P n-1 n-1 c(c+1) x-2

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19 Analysis - Cycle No more than c vertices with maximum rank Remove those vertices with maximum rank will result no more than c paths At least one of the path must have no less than (n-c)/c vertices that path needs at least ranks ranks The whole cycle needs at least ranks

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20 Theorem The c-vertex ranking number of wheel W n is

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21 Wheel (W n =K 1 +C n ) 1 1 1 1 1 1 2 2 3 3 If rank of v c (say x) is the same as max rank in path P n-1 then

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22 Wheel (W n =K 1 +C n ) 1 1 1 1 1 1 2 2 3 3 If rank of v c (say x) is one more than max rank in path P n-1 then there are at least c-1 vertices ranked as x-1 in the path

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23 Analysis - Wheel Case 1: v c is the only vertex with max rank Since

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24 Analysis - Wheel Case 2: v c is not the only vertex with max rank The graph after removing the vertices with max rank is a collection (no more than c-1) of paths There is a path with at least vertices

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25 Analysis - Wheel Case 3: v c is not ranked with max rank (say x) Remove the vertices with max rank won ’ t separate the graph Assume remove vertices with rank greater than x-y will separate the graph to no more than cy-1 paths There exists a path containing at least vertices

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26 Theorem The c-vertex ranking number of complete bipartite graph K m,n for m n is

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27 Complete Bipartite Graph all vertices can be ranked as 1 All vertices in m partite set ranked as 1 The vertices in n partite set ranked as 2 to n m

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28 Corollary The c-vertex ranking number of complete r- partite graph for is

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29 Thank you

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