1 / 24

Towards an algebra of connectivity

For too long has the structural graph theory been exploring flavors of connectivities without a clear subsuming approach. This is not it. But it's something more in that direction and has been far from barren, so far.

thorwhalen1
Download Presentation

Towards an algebra of connectivity

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Subdivisibility Towards an algebra of connectivity Thor Whalen, Mathematician, August 2003

  2. H-subdivisions G H 1 2 1 2 4 3 4 3

  3. H-linked G H 1 2 1 2 4 3 4 3

  4. H-linked G H 3 2 1 2 1 4 4 3

  5. H-linked G 1 4 2 H 3 2 1 3 2 1 2 3 1 4 4 3 4

  6. H-linked: A generalization of connectivity H-linked means: H 3-connected 3-connected 3-connected 3-connected The H-linked concept can express: k-orderedness “k-linkedness” k-connectivity 2-linked etc… Ubiquity of cycles with prescribed and proscribed vertices Ubiquity of linear-forests 4-ordered

  7. H-linked: A generalization of connectivity H-linked means: H 3-connected 3-connected 3-connected 3-connected The H-linked concept can express: k-orderedness “k-linkedness” k-connectivity 2-linked etc… Ubiquity of cycles with prescribed and proscribed vertices Ubiquity of linear-forests 4-ordered

  8. H-linked: A generalization of connectivity H-linked means: H 3-connected 3-connected 3-connected 3-connected The H-linked concept can express: k-orderedness “k-linkedness” k-connectivity 2-linked etc… Ubiquity of cycles with prescribed and proscribed vertices Ubiquity of linear-forests 4-ordered

  9. H1→ H2 A E N |G|≥N, G is H1-linked G is H2-linked. def H1is stronger than H2 Connectivity Strength 12k- 16k- 10k- 10k-

  10. H2subgraph of H1 H1→ H2 Connectivity StrengthPoset Obvious fact: These all “mean” 3-connected!

  11. : G→R μ Connectivity StrengthPoset μ-threshold for H1-linked There are H2-linked graphs that are not H1-linked X μ-threshold for H2-linked X

  12. : G→R μ Connectivity StrengthPoset μ-threshold for H1-linked There are H2-linked graphs that are not H1-linked X μ-threshold for H2-linked X 4 1 2 x n-x-2-4-1 μ(G) = x

  13. 4 -linked 1 2 Yes! x n-x-2-4-1 μ(G) = x Connectivity StrengthPoset μ-threshold for H1-linked X μ-threshold for H2-linked X

  14. -linked No! Connectivity StrengthPoset μ-threshold for H1-linked X μ-threshold for H2-linked X -linked Yes!

  15. Theorem: If |H1|=|H2|, with exception to subgraphs of a star; H1→H2 H2subgraph of H1 Connectivity StrengthPoset

  16. customary property name min-degree threshold H n+k-2 k-connected 2 n+k/2-2 k-ordered 2 n+2k-3 k-linked 2 etc... Min-degree threshold for H-linkage For connected H, η(H) is the order of a maximal edge cut η(H) = 3 Ferrara, Gould, Tansey, Whalen n+η(H)-2 H-linked 2 Kostochka, Yu

  17. 1 2 4 3 H-extendibility 1 A graph G is H-extendible if whenever G has an H-subdivision on S… 2 4 3

  18. 1 2 4 3 H-extendibility 1 A graph G is H-extendible if whenever G has an H-subdivision on S… 2 … it has a spanning H-subdivision. 4 3

  19. H-extendible expresses the “spanning property” in concepts such as… Suff. cond. implying extendibility H Hamiltonicity δ(G) ≥ n/2 Hamiltonian connectivity δ(G) ≥ (n+1)/2 Hamiltonian k-ordered К(G) ≥ 1+k/2 δ(G) ≥ n/2 Hamiltonian k-linked К(G) ≥ k+1 δ(G) ≥ (n+k)/2 etc... Min-degree threshold for H-extendibility A graph G is H-extendible if whenever G has an H-subdivision on S… … it has a spanning H-subdivision. Gould, Whalen Where… α(H) = vertex indep. number β(H) = edge indep. number ξ(H) = ρ(H)-1+ h1(H) + 2h0(H) (“rank” ρ(H) = |E(H)| - |H| + 1) К(G) ≥ 1+max(α(H), β(H)) δ(G) ≥ (n+ξ(H))/2 H-extendible

  20. Further Research “H-pan-extendibility” (subdivisions of all sizes) Gould, Magnant, Powell, Wagner, Whalen “H-reducibility” (small subdivisions) Ore degree for H-linked graphs • More theorems about H-linked strength • Other sufficient conditions for H-linked graphs • Develop an appropriate “H-language” to encompass other graphical properties

  21. { SH(G) = (1, 2, 3, 4), (2, 1, 3, 4), (2, 1, 3, 5), (2, 1, 4, 5), (2, 2, 4, 5), (3, 1, 4, 5), (3, 1, 4, 5), (3, 1, 4, 5), (4, 1, 3, 5), (4, 2, 5, 6), (5, 1, 4, 6), (5, 2, 3, 6), (6, 2, 3, 4), etc. -linked means 3 2 1 2 3 4 3 4 6 5 1 2 min degree at least 3 -linked means 3-connected and min degree at least 3 6 5 4 3 2 1 -linked means NOT graph is a forest } Gauging Linkage Fixed Vertices Free Vertices ,F) (H G “Neither (K5,Ø)-linked nor (K3,3,Ø)-linked” means graph is planar. {1,2,3,4,5,6} H

  22. And now….Chalkboard discussions…

  23. 1 2 3 4 5 6

More Related