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

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)

3 C 8 is 5-rankable;  r (C 8 )=

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

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

6 Example ( On-line) T Off-line T On-line v1v1 v2v2 v7v7 v4v4 v8v8 v3v3 v5v5 v6v6 n=8

7 Bounds in online version  

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

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)

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

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

12 Example (separator tree ) T

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

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

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

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

17 Cycle 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 

18 Cycle 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 

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

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

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

22 Wheel (W n =K 1 +C n )  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 

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

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  

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  

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

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

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

29 Thank you