unavoidable induced subgraphs of large 2 connected graphs
play

Unavoidable Induced Subgraphs of Large 2-Connected Graphs Sarah - PowerPoint PPT Presentation

Mar 24, 2024 •153 likes •721 views

Unavoidable Induced Subgraphs of Large 2-Connected Graphs Sarah Allred* Guoli Ding Bogdan Oporowski Louisiana State University 31st Cumberland Conference on Combinatorics, Graph Theory and Computing Background Theorem (Ramsey (1928)) There

  1. Unavoidable Induced Subgraphs of Large 2-Connected Graphs Sarah Allred* Guoli Ding Bogdan Oporowski Louisiana State University 31st Cumberland Conference on Combinatorics, Graph Theory and Computing

  2. Background Theorem (Ramsey (1928)) There is a function f such that every graph on f ( r ) vertices contains an induced K r or K r . Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  3. Background Theorem (Ramsey (1928)) There is a function f such that every graph on f ( r ) vertices contains an induced K r or K r . Theorem (Ramsey (1928)) Every giant graph contains a large induced K r or K r Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  4. Background Theorem (Ramsey (1928)) Every giant graph contains a large induced K r or K r . Theorem Every giant connected graph contains a large induced K r , K 1 , r , or P r . Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  5. Background Theorem (Ramsey (1928)) Every giant graph contains a large induced K r or K r . Theorem Every giant connected graph contains a large induced K r , K 1 , r , or P r . Theorem Every giant 2 -connected graph contains a subdivision of K 2 , r or C r . Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  6. Main Result What about 2-connected induced subgraphs? Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  7. Main Result Theorem (A., D.,O. 2019+) Every gargantuan 2 -connected graph contains a large K r , TK 2 , r , TK + 2 , r , or DL r as an induced subgraph. Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  8. Main Result Theorem (A., D.,O. 2019+) Every gargantuan 2 -connected graph contains a large K r , TK 2 , r , TK + 2 , r , or DL r as an induced subgraph. TK + K r TK 2 , r 2 , r DL r Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  9. Main Result Theorem (A., D.,O. 2019+) Every gargantuan 2 -connected graph contains a large K r , TK 2 , r , TK + 2 , r , or DL r as an induced subgraph. TK + K r TK 2 , r 2 , r DL r Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  10. Main Result Theorem (A., D.,O. 2019+) Every gargantuan 2 -connected graph contains a large K r , TK 2 , r , TK + 2 , r , or DL r as an induced subgraph. TK + K r TK 2 , r 2 , r DL r Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  11. Graphs without a long path Lemma (A.,D.,O.,2019+) Every giant 2-connected graph contains either a long path or an induced TK 2 , r or TK + 2 , r . Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  12. Graphs without a long path Lemma (A.,D.,O.,2019+) Every giant 2-connected graph contains either a long path or an induced TK 2 , r or TK + 2 , r . TK + TK 2 , r 2 , r Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  13. Graphs without a long path TK + TK 2 , r 2 , r Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  14. Graphs without a long path TK + TK 2 , r 2 , r Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  15. Graphs without a long path TK + TK 2 , r 2 , r Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  16. Graphs without a long path TK + TK 2 , r 2 , r Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  17. Graphs without a long path TK + TK 2 , r 2 , r Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  18. Graphs without a long path TK + TK + TK 2 , r 2 , r 2 , r Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  19. Graphs without a long path TK + TK 2 , r 2 , r Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  20. Graphs without a long path TK + TK 2 , r TK 2 , r 2 , r Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  21. Graphs with long path Theorem (Galvin, Rival, Sands) Every graph with a really long path contains a large induced K r , K r , r , or P r . Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  22. Graphs with long path Theorem (Galvin, Rival, Sands) Every graph with a really long path contains a large induced K r , K r , r , or P r . Lemma (A., D.,O. 2019+) Every 2 -connected graph with a really long path contains a large induced K r , K r , r , or long messy ladder. Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  23. Graphs with long path Theorem (Galvin, Rival, Sands) Every graph with a really long path contains a large induced K r , K r , r , or P r . Lemma (A., D.,O. 2019+) Every 2 -connected graph with a really long path contains a large induced K r , K r , r , or long messy ladder. Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  24. Tutte Bridges Tutte Bridges Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  25. Tutte Bridges Tutte Bridges Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  26. Tutte Bridges Tutte Bridges Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  27. Tutte Bridges Tutte Bridges Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  28. Long Path to Messy Ladder Q 1 Q 2 Q 3 P Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  29. Long Path to Messy Ladder Q 1 Q 2 Q 3 P Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  30. Long Path to Messy Ladder Q 1 Q 2 Q 3 P Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  31. Long Path to Messy Ladder Q 1 Q 1 Q 2 Q 3 P Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  32. Long Path to Messy Ladder Q 1 Q 2 Q 3 P Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  33. Long Path to Messy Ladder Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  34. Long Path to Messy Ladder Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  35. Long Path to Messy Ladder Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  36. Messy Ladder to Deformed Ladder C r K r TK + TK 2 , r F r 2 , r Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  37. Messy Ladder to Deformed Ladder C r K r TK + TK 2 , r F r 2 , r Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  38. Messy Ladder to Deformed Ladder C r K r TK + TK 2 , r F r F r 2 , r Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  39. Messy Ladder to Deformed Ladder C r K r TK + TK 2 , r F r 2 , r Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  40. Messy Ladder to Deformed Ladder C r C r K r TK + TK 2 , r F r 2 , r Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  41. Messy Ladder to Deformed Ladder K r TK + C r TK 2 , r F r 2 , r Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  42. Messy Ladder to Deformed Ladder C r C r K r TK + TK 2 , r F r 2 , r Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  43. Messy Ladder to Deformed Ladder After a series of other observations we get... Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  44. Messy Ladder to Deformed Ladder C r K r TK + TK 2 , r F r 2 , r Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  45. Messy Ladder to Deformed Ladder C r K r TK + TK 2 , r F r 2 , r Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  46. Messy Ladder to Deformed Ladder C r K r TK + TK 2 , r F r 2 , r Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  47. Messy Ladder to Deformed Ladder K r TK + C r TK 2 , r F r 2 , r Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

Recommend

Planar Induced Subgraphs of Sparse Graphs Glencora Borradaile, David Eppstein , and Pingan Zhu

Planar Induced Subgraphs of Sparse Graphs Glencora Borradaile, David Eppstein , and Pingan Zhu

Planar Induced Subgraphs of Sparse Graphs Glencora Borradaile, David Eppstein , and Pingan Zhu Graph Drawing 2014 The planarization problem Goal: find big planar subgraphs in nonplanar graphs Equivalently: delete as little as possible so the

397 views • 17 slides
CONFIGURATIONS IN LARGE t -CONNECTED GRAPHS Robin Thomas School of Mathematics Georgia Institute

CONFIGURATIONS IN LARGE t -CONNECTED GRAPHS Robin Thomas School of Mathematics Georgia Institute

CONFIGURATIONS IN LARGE t -CONNECTED GRAPHS Robin Thomas School of Mathematics Georgia Institute of Technology http://math.gatech.edu/~thomas joint work with Sergey Norin A minor of G is obtained by taking subgraphs and contracting edges.

607 views • 42 slides
Decomposing Cubic Graphs into Connected Subgraphs of Size Three Laurent Bulteau Guillaume Fertin

Decomposing Cubic Graphs into Connected Subgraphs of Size Three Laurent Bulteau Guillaume Fertin

Decomposing Cubic Graphs into Connected Subgraphs of Size Three Laurent Bulteau Guillaume Fertin Anthony Labarre Romeo Rizzi Irena Rusu March 24, 2017 The graph decomposition problem Given a set S of graphs, an S -decomposition of a graph G

850 views • 66 slides
Connected subgraphs in edge-coloured graphs Henry Liu 1 Based on a joint survey with Shinya Fujita

Connected subgraphs in edge-coloured graphs Henry Liu 1 Based on a joint survey with Shinya Fujita

Connected subgraphs in edge-coloured graphs Henry Liu 1 Based on a joint survey with Shinya Fujita 2 and Colton Magnant 3 1 New University Lisbon, Portugal 2 Yokohama City University, Japan 3 Georgia Southern University, USA Discrete Mathematics

1.14k views • 87 slides
Computing Maximum Unavoidable Subgraphs Using SAT Solvers Cuong Chau & Marijn Heule {

Computing Maximum Unavoidable Subgraphs Using SAT Solvers Cuong Chau & Marijn Heule {

Computing Maximum Unavoidable Subgraphs Using SAT Solvers Cuong Chau & Marijn Heule { ckcuong,marijn } @cs.utexas.edu Department of Computer Science The University of Texas at Austin July 7, 2016 Cuong Chau & Marijn Heule (UT Austin)

515 views • 47 slides
K t MINORS IN LARGE t -CONNECTED GRAPHS Robin Thomas School of Mathematics Georgia Institute of

K t MINORS IN LARGE t -CONNECTED GRAPHS Robin Thomas School of Mathematics Georgia Institute of

K t MINORS IN LARGE t -CONNECTED GRAPHS Robin Thomas School of Mathematics Georgia Institute of Technology http://math.gatech.edu/~thomas joint work with Sergey Norin K t MINORS IN LARGE t -CONNECTED GRAPHS Robin Thomas School of Mathematics

658 views • 54 slides
Colouring graphs excluding fixed subgraphs joint work with S. Thomass, M. Bonamy Problem Very

Colouring graphs excluding fixed subgraphs joint work with S. Thomass, M. Bonamy Problem Very

Colouring graphs excluding fixed subgraphs joint work with S. Thomass, M. Bonamy Problem Very General Question : What does having large chromatic number say about a graph? Problem Very General Question : What does having large chromatic

809 views • 54 slides
On the Size and the Approximability of Minimum Temporally Connected Subgraphs Dimitris Fotakis

On the Size and the Approximability of Minimum Temporally Connected Subgraphs Dimitris Fotakis

On the Size and the Approximability of Minimum Temporally Connected Subgraphs Dimitris Fotakis Yahoo! Research, New York National Technical University of Athens Joint work with Kyriakos Axiotis , CSAIL, MIT NYCAC, November 2017 Dimitris

798 views • 49 slides
Sierpi nski graphs as spanning subgraphs of Hanoi graphs Sara Sabrina Zemlji c Andreas M.

Sierpi nski graphs as spanning subgraphs of Hanoi graphs Sara Sabrina Zemlji c Andreas M.

Definitions Main result Sierpi nski graphs as spanning subgraphs of Hanoi graphs Sara Sabrina Zemlji c Andreas M. Hinz Sandi Klav zar Institute of Mathematics, Physics and Mechanics September 2012 Definitions Main result Hanoi

448 views • 12 slides
Polynomial Chi-binding functions and forbidden induced subgraphs a survey Ingo Schiermeyer TU

Polynomial Chi-binding functions and forbidden induced subgraphs a survey Ingo Schiermeyer TU

Polynomial Chi-binding functions and forbidden induced subgraphs a survey Ingo Schiermeyer TU Bergakademie Freiberg Chromatic Number Ingo Schiermeyer GGTW 2019 Chromatic Number Andrs Gyrfs (Budapest, 1985) Problems from the world

337 views • 29 slides
Colorful complete bipartite subgraphs in generalized Kneser graphs Fr ed eric Meunier

Colorful complete bipartite subgraphs in generalized Kneser graphs Fr ed eric Meunier

Colorful complete bipartite subgraphs in generalized Kneser graphs Fr ed eric Meunier August 30th, 2017 MSRI Seminar Joint work with Meysam Alishahi and Hossein Hajiabolhassan Any proper 3 -coloring of the Petersen graph contains a C 6

586 views • 36 slides
The independence numbers and the chromatic numbers of random subgraphs of Knesers graphs and

The independence numbers and the chromatic numbers of random subgraphs of Knesers graphs and

The independence numbers and the chromatic numbers of random subgraphs of Knesers graphs and their generalizations Andrei Raigorodskii Moscow Institute of Physics and Technology, Yandex, Moscow, Russia Shanghai, China, 26.03.2017 A.

837 views • 56 slides
Faithful subgraphs and Hamiltonian circles of infinite graphs Binlong Li Northwestern

Faithful subgraphs and Hamiltonian circles of infinite graphs Binlong Li Northwestern

Faithful subgraphs and Hamiltonian circles of infinite graphs Binlong Li Northwestern Polytechnical University Ghent, August, 2019 Outline Basic terminology Main result Applications of the main result Locally finite; Ray; Equivalent; End G

294 views • 17 slides
k -Connected Graphs 1 We can now further extend a few of the concepts we discussed with

k -Connected Graphs 1 We can now further extend a few of the concepts we discussed with

k -Connected Graphs 1 We can now further extend a few of the concepts we discussed with restriction to 2- connected and 2-edge-connected to k -connected and k -edge-connected graphs. Given two vertices x, y V ( G ), a set S V ( G ) is an

177 views • 3 slides
Graphs Graphs What is connected to what Many things we deal with in computer science are

Graphs Graphs What is connected to what Many things we deal with in computer science are

Graphs Graphs What is connected to what Many things we deal with in computer science are graphs Networks: humans, communication, computation, transportation, knowledge Courtesy:gigaflop.demon.co.uk Courtesy: cablemap.info Courtesy:

293 views • 8 slides
Spanning F -free subgraphs with large minimum degree Guillem Perarnau SIAM Conference on Discrete

Spanning F -free subgraphs with large minimum degree Guillem Perarnau SIAM Conference on Discrete

Spanning F -free subgraphs with large minimum degree Guillem Perarnau SIAM Conference on Discrete Math, Minneapolis - June 19th, 2014 McGill University, Montreal, Canada joint work with Bruce Reed. The problem Let G be a large graph and F a

747 views • 37 slides
Kernel lower bounds using co-nondeterminism: Finding induced hereditary subgraphs Stefan Kratsch,

Kernel lower bounds using co-nondeterminism: Finding induced hereditary subgraphs Stefan Kratsch,

Kernel lower bounds using co-nondeterminism: Finding induced hereditary subgraphs Stefan Kratsch, Marcin Pilipczuk, Ashutosh Rai, Venkatesh Raman SWAT 2012, Helsinki S. Kratsch, M. Pilipczuk, A. Rai, V. Raman Kernel lower bounds using

1.32k views • 75 slides
Graphs () Graphs () Graphs Graphs Graphs are collections of nodes

Graphs () Graphs () Graphs Graphs Graphs are collections of nodes

Graphs () Graphs () Graphs Graphs Graphs are collections of nodes in which various pairs are connected by K08 line segments. The

627 views • 17 slides
Graphs Graphs Examples Definitions Implementation/Representation of graphs Graphs

Graphs Graphs Examples Definitions Implementation/Representation of graphs Graphs

CS171 Introduction to Computer Science II Science II Graphs Graphs Examples Definitions Implementation/Representation of graphs Graphs Graphs: set of vertices connected pairwise by edges Interesting and useful structure

346 views • 30 slides
Coalition games on interaction graphs Nicolas Bousquet joint work with Zhentao Li and Adrian

Coalition games on interaction graphs Nicolas Bousquet joint work with Zhentao Li and Adrian

Coalition games on interaction graphs Nicolas Bousquet joint work with Zhentao Li and Adrian Vetta Nyborg, August 2018 1/15 The problem Let G = ( V , E ) be a graph. Let C be a collection of connected subgraphs of G . Maximum Packing : Maximum

514 views • 39 slides
Thresholds in random graphs with focus on thresholds for k -regular subgraphs Pawe Praat

Thresholds in random graphs with focus on thresholds for k -regular subgraphs Pawe Praat

Model and Thresholds Lower Bound Upper Bound Some details Thresholds in random graphs with focus on thresholds for k -regular subgraphs Pawe Praat Department of Mathematics, Ryerson University, Toronto, ON, Canada Pawe Praat k

1.01k views • 68 slides
Graphs Graphs Simple graphs Algorithms Depth-first search Breadth-first search

Graphs Graphs Simple graphs Algorithms Depth-first search Breadth-first search

CS171 Introduction to Computer Science II Graphs Graphs Simple graphs Algorithms Depth-first search Breadth-first search shortest path Connected components Directed graphs Weighted graphs Minimum spanning tree

781 views • 53 slides
An introduction to ribbon graphs Iain Moffatt Royal Holloway, University of London ALEA-Network

An introduction to ribbon graphs Iain Moffatt Royal Holloway, University of London ALEA-Network

An introduction to ribbon graphs Iain Moffatt Royal Holloway, University of London ALEA-Network Workshop Bordeaux, 16-18 November 2015 Contents What is a ribbon graph? What is a ribbon graph? Subgraphs and minors Polynomials Subgraphs

896 views • 60 slides
The Mean Subtree Order and the Mean Connected Induced Subgraph Order Lucas Mol Joint work with

The Mean Subtree Order and the Mean Connected Induced Subgraph Order Lucas Mol Joint work with

B ACKGROUND T HE GLUING LEMMA B EYOND T REES O UTLOOK The Mean Subtree Order and the Mean Connected Induced Subgraph Order Lucas Mol Joint work with Kristaps Balodis and Ortrud Oellermann (The University of Winnipeg), and Matthew Kroeker (The

1.65k views • 132 slides

More recommend