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Bandwidth Sum Problem

Bandwidth Sum Problem. Jing-Ho Yan( 顏經和 ) Department of Mathematics Aletheia University ( 真理大學 ). Introduction (1). A k - coloring of a graph G is a labeling f:V ( G )  S , where | S |= k (or S= [ k ]={1,2, … , k } ) and adjacent vertices have different colors.

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Bandwidth Sum Problem

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  1. Bandwidth Sum Problem Jing-Ho Yan(顏經和) Department of Mathematics Aletheia University (真理大學)

  2. Introduction (1) • A k-coloringof a graph G is a labeling f:V(G)S, where |S|=k(or S=[k]={1,2,…,k}) and adjacent vertices have different colors. • A graph G has a k-coloringiff it has a vertex mapping f:V(G)  V(Kk)such that for any two vertices u,vV(G), uvE(G)  f(u)f(v)

  3. Introduction (2) • A k-coloringof a graph G is a labeling f:V(G)S, where |S|=k(or S=[k]={1,2,…,k}) and adjacent vertices have different colors. • A graph G has a k-coloringiff it has a vertex mapping f:V(G)  V(Kk)such that for any two vertices u,vV(G), uvE(G)  d( f(u), f(v) ) =1

  4. Introduction (3) • A k-coloringof a graph G is a labeling f:V(G)S, where |S|=k(or S=[k]={1,2,…,k}) and adjacent vertices have different colors. • A graph G has a k-coloringiff it has a vertex mapping f:V(G)  V(Pk)such that for any two vertices u,vV(G), uvE(G)  d( f(u), f(v) )1

  5. Introduction (4) • A mapping f:V(G)  V(Pk)such that for any two vertices u,vV(G), uvE(G)  d( f(u), f(v) )  p • A mapping f :V(G){0,1,2,…,k-1}such that |f(u)-f(v)|p if uvE(G) (k-1)-L(p,0)-labeling

  6. Introduction (5) A mapping f:V(G)  V(Pk)such that for any two vertices u,vV(G), d(u,v)=1  d( f(u), f(v) )  p d(u,v)=2  d( f(u), f(v) )  q A mapping f :V(G){0,1,2,…,k-1}such that |f(u)-f(v)|p if d(u,v)=1 |f(u)-f(v)|q if d(u,v)=2 (k-1)-L(p,q)-labeling

  7. Introduction (6) A mapping f:V(G)  V(Pk)such that for distinct vertices u,vV(G), we have d(u,v) + d( f(u), f(v) ) > 2 A mapping f :V(G){0,1,2,…,k-1}such that |f(u)-f(v)|2 if d(u,v)=1 |f(u)-f(v)|1if d(u,v)=2 (k-1)-L(2,1) labeling

  8. Introduction (6) A mapping f:V(G)  V(Pk)such that for distinct vertices u,vV(G), we have d(u,v) + d( f(u), f(v) ) > 2 A mapping f :V(G){0,1,2,…,k-1}such that d(u,v) +|f(u)-f(v)| > 2 (k-1)-L(2,1) labeling

  9. Introduction (7) A mapping f:V(G)  V(Pk)such that for distinct vertices u,vV(G), we have d(u,v) + d( f(u), f(v) ) > daim(G) A mapping f :V(G){0,1,2,…,k-1}such that d(u,v) +|f(u)-f(v)| > daim(G) (k-1)-radio labeling

  10. Introduction (8) An 1-1 mapping f:V(G)  V(Pk)such that for any two vertices u,vV(G), uvE(G)  d( f(u), f(v) )  p An 1-1 mapping f :V(G){0,1,2,…,k-1}such that |f(u)-f(v)|p if d(u,v)=1 (k-1)-L’(p,1)-labeling

  11. Introduction (9) An 1-1 mapping f:V(G)  V(Pk)such that for any two vertices u,vV(G), uvE(G)  d( f(u), f(v) )  p An 1-1 mapping f :V(G){0,1,2,…,k-1}such that |f(u)-f(v)|  p if uvE(G) ?? - labeling

  12. Introduction (10) An 1-1 mapping f:V(G)  V(P|V(G)|)such that for any two vertices u,vV(G), uvE(G)  d( f(u), f(v) )  p The bandwidth Bf(G)of Gwith respect tofis minp. That is,

  13. F.R.K. Chung [1988] • Label the vertices of a graph G by distinct vertices of a 'host' graphH, subject to one of the following conditions. • (1) The maximum distance in H between adjacent vertices in G is minimized. • (2) The total sum of distances in H between adjacent vertices in G is minimized.

  14. Suppose G is a graph of order n. Then we have the following relations. bandwidth  (1) + host graphPn. cyclic bandwidth (1) + host graphCn. bandwidth sum (2) + host graphPn. cyclic bandwidth sum (2) + host graphCn.

  15. Definition A proper labelingf of G is a one-to-one mapping f :V(G){1,2,,|V(G)|}. Supposefis a proper labeling of G. The bandwidth Bf (G)( ) of Gwith respect tofis defined by bandwidth sum BSf (G)

  16. 4 1 3 1 1 1 Example 1 Bf (G)=4 2 BSf (G)=11 5 3 4

  17. The bandwidth B(G)(bandwidth sum BS(G)) of G is defined by B(G)= min{Bf (G)| f is a proper labeling of G}. BS(G)= min{BSf (G)| f is a proper labeling of G}. A bs-optimal labeling of G is a proper labeling f and BSf (G) = BS(G).

  18. Example 4 3 1 2 3 4 1 2

  19. Question: Given a symmetric matrix M , can we find a symmetric permutation M’ of M, so that the number is minimum? Find B(G), where G is the graph model M.

  20. SupposeS is a vertex subset of G. is an edge cut. • Let (S) = | | • Then

  21. Example 1 Bf (G)=4 BSf (G)=11 2 5 3 4

  22. Supposefis a proper labeling of G. • The profile P(G)of G is defined by P(G)= min{Pf (G)| f is a proper labeling of G}. • In fact,

  23. 4 1 3 1 1 1 Example 1 Bf (G)=4 2 BSf (G)=11 5 Pf (G)=0+1+1+1+4=6 3 4

  24. union The union of two graph G and H, denoted by GH, is a graph with

  25. The join of two graph G and H, denoted by G+H, is a graph with join

  26. Cograph • Cographs are defined recursively by the following rules: • A trivial graph is a cograph. • If G and H are cographs, then the join G+Hof Gand H is a cograph. • If G and H are cographs, then the union GH of G and H is a cograph.

  27. Is it NP-complete when we known BS(G), BS(H)? Is it polynomial when we known BS(G), BS(H)??

  28. Known Results • In 1994, Lai and Williams present a polynomial time algorithm to establish BS(G) for G = G1 + G2 + · · · + Gk with each Gisum deterministic.

  29. Sum Deterministic • A proper D-labelingf of G is a one-to-one mapping f : V(G) D, where D is an integer subset and |D|=|V(G) |. • D-bandwidth sum BS(G;D) = min{BSf (G;D)| f is a proper D-labeling of G}. • A graph G of order n is called sum deterministic if for each positive integer set D={p1, p2, · · ·, pk} with p1<p2 < · · · < pk, BSf (G)=BS(G) if and only if the labeling g, defined by g(v)=pf(v) for each vertex v in G, satisfies BSf (G;D)=BS(G;D) .

  30. Sum Deterministic D={1,3,4} Not Sum Deterministic 4 1 1 1 1 3 2 3 2 3 1 3 4 3 4

  31. Known Results • (2005) Chen, Kuo, and Yan For the graph G = G1 + G2 + · · · + Gk, where Gi is a path (with at least two vertices), cycle, or union of isolated vertices, letn = |V (G)|,pi=|V(Gi)| − (Gi ). If p1p2 · · · pk, then BS(G)=BS(H1+G2+···+Gk)+(n−|V(G1)|+(G1))(n−1), where H1 = (|V(G1)|−2)K1.

  32. In General

  33. Known Results • (2005) Chen, Kuo, and Yan Let G = G1 + G2 + · · · + Gk, where Gi is a path, cycle, or union of isolated vertices. Then

  34. Corollary Liu and Williams (1995)

  35. Theorem • Chang, Chia, Kuo, Lin, and Yan

  36. Quasi-Threshold Graphs • Cographs are defined recursively by the following rules: • A trivial graph is a cograph. • If G and H are cographs, then the join G+Hof Gand H is a cograph. • If G and H are cographs, then the union GH of G and H is a cograph. • In 2., if G=K1, then we can define Quasi-threshold graphs.

  37. Quasi-threshold graph • The induced graph of a rooted forest F = (V,E) is the graph G(F) =(V,E0), where E0={ uv | directed u-v or v-u path in F}. • A graph G is a quasi-threshold graph if and only if G is induced by a rooted forest. • The graph K1 + Kni is the quasi-threshold graph induced by a rooted tree in which each vertex has at most one child except the root.

  38. Join of K1and G • Let G be a graph of order n. Then , where is an edge cut of G. • If edge cut is minimum, then

  39. Join of K1and G • Suppose G is a the quasi-threshold graph induced by a complete m-ary tree of height h. Then

  40. Threshold Graphs Cographs are defined recursively by the following rules: A trivial graph is a cograph. If G and H are cographs, then the join G+Hof Gand H is a cograph. If G and H are cographs, then the union GH of G and H is a cograph. In 2 and 3, if G=K1, then we can define Threshold graphs.

  41. Threshold Graphs A connected threshold graph is an induced graph of a rooted tree T, where T - { u | u is a leaf in T } is a directed path. We give a polynomial-time algorithm to compute bandwidth sum of a threshold graph by a recurrence relation on subtree of T.

  42. Let (i)=min{(S) | Sis a vertex k-subset of G.} Supposefis a proper labeling of G. We let f(i)=|f -1({1,2,…,i})|. We say f complete if f (i)=(i)for all i. Theorem: If G has a complete labeling f, then f is bs-optimal labeling and BS(G)=(i). Complete labeling

  43. Join of K1and G Suppose G is a graph of order n and has a complete labeling f. Then

  44. Join of K1and G Suppose G is a graph of order n and has a bs-optimal labeling f such that f (n/2)=(n/2). If |S|(S) for each SV(G) and |S|n/2, then Corollary:

  45. Join of two graphs Suppose G is a graph of order n and has a complete labeling f. If (i)-(i-1)n-2ifor all 2in/2, then

  46. Join of two hypercubes If mn, then

  47. Corona • The corona of G and H is the graph GH with V(GH)=V(G)∪(V(G)×V(H)) E(GH)=E(G)∪{u(u,v) | u∈V(G) and v∈V(H)} ∪{(u,v)(u,w) | u∈V(G) and vw∈E(H)}.

  48. Theorem (Chia, Kuo, Huang, and Yan) • If m≤2n and m is even, then • If m≤2n and m is odd, then • If m>2n and m is even, then • If 2n<m3n+1 and m is odd, then • If m>3n+1, m is odd, and n is even, then • If m>3n+1 and mn is odd, then

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