1. 國立東華大學應用數學系 林 興 慶 Lin-Shing-Ching 指導教授 : 郭大衛 Vertex Ranking number of Graphs 圖的點排序數

2. A vertex ranking of a graph G is a mapping f from V(G) to the set of all natural number such that for any path between two distinct vertices u and v with f(u)=f(v) there is a vertex w in the path f(w)>f(u).In this definition,we call the value f(v) the rank of the vertex v.

3. The vertex ranking problem is to determine the vertex ranking number r(G) of a given graph G. 135792846 111112234 r(G)=4

4. Lemma If H is a subgraph of G,then r(H) ≦ r(G).

5. The union of two disjoint graph G and H is the G ∪ H vertex set V(G ∪ H)=V(G) ∪ V(H) edge set E(G ∪ H)=E(G) ∪ E(H) Lemma

6. 111112234 11123 r(G 1 )=4 r(G 2 )=3 r(G 1 ∪ G 2 )=4

7. 113112234 1 1 1 1 1 1 1 1 12 2 2 3 3 4 45 6 P9P9 ×P2×9×P2×9

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

9. 1 P 12 2234 G˙S 11111 32 P 12 111 222223345 G˙S

10. Lemma For any graph G, r(G)=min{r(G˙S)+1:S is an independent set of G}

11. The join of two graph G and H is the G + H with vertex set V(G+H)=V(G) ∪ V(H) edge set E(G+H)=E(G) ∪ E(H) ∪ {xy:x in G,y in H} Lemma

12. r(G)=4 r(G+H)=r(G)+ r(G+H)=r(G)+ |V(H)|=6 11 3 11223 4 P9P9 P2P2

13. G r is a graph with V(G r )=V(G) E(G r )={uv ｜ u,v ∈ V(G) and d G (u,v) ≦ r} P72P72 1122345 r(P 7 2 )=5

14. P 30 2

16. Theorem For all n,k with n ≧ 3 and n ≧ k-1

17. Theorem For all Theorem For all

18. Cartesian product of two graph G and H is G×H V(G×H)={(u,v) ｜ u ∈ V(G) v ∈ V(H)} E(G×H)={(u,x)(v,y) ｜ (u=v),xy ∈ E(H) or(uv ∈ E(G),x=y)}

19. Try to P 2 × n ×P2×9×P2×9 r(P 2 × 9 )=6 1 1 1 11 1 1 1 2 22 23 4 3 4 5 6

20. ×2k+1 P 2 ×2k+1 ×2k P 2 ×2k

22. Theorem For a caterpillar T. Let P n be the subgraph of T obtained from T by deleting all leaves of T,and {v ∈ V(P n ):d G ≧ 3 }={v j1,v j2,…,v jk }, where j 1 ≦ j 2 ≦ … ≦ j k. If we let j 0 =0,j k+1 =n+1. then r(T)=r(P l )+1 where

23. 11 1 1 1 1 1 11 1 1 v1v2 v3v4v5v6 11111111 2225234223334

24. Composition of two graph,written G[H] vertex set V(G) ×V(H) edge set (u 1,v 1 )is adjacent to(u 2,v 2 ) if eithher u 1 is adjacent to u 2 in G G[H]=P 4 [P 5 ] r(G[H])=r(P 4 [P 5 ])=r(P 4 )+ 2|V(P 5 )|=11 1 2 3 1 4 5 6 7 1 2 3 1 4 5 6 7 8 9 10 11

25. Theorem For any two graphs G and H.

26. vertex set edge set

27. V 1,1 V 2,1 V 3,1 r(H 1 )=lr(H 2 )=lr(H 3 )=l

28. 1123 11223 4 1 1 2 3 4 1123 5 12 3 6

29. Corona of two graphs,written G Λ H. vertex set V(GΛH)=V(G) ∪ {u ij |1 ≦ i ≦ n,1 ≦ j ≦ m} edge set Corona of two graphs,written G Λ H. vertex set V(GΛH)=V(G) ∪ {u ij |1 ≦ i ≦ n,1 ≦ j ≦ m} edge set

30. 211211211211211 11223 Theorem For any two graphs G and H. 33445

31. 感謝各位的參與