local properties of graphs and the hamilton cycle problem
play

Local Properties of Graphs and the Hamilton Cycle Problem Johan de - PowerPoint PPT Presentation

Apr 28, 2023 •106 likes •1.26k views

Local Properties of Graphs and the Hamilton Cycle Problem Johan de Wet 1 , 2 and Marietjie Frick 1 1 University of Pretoria 2 DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS) Bucharest 2018 Johan de Wet et al. (UP,

  1. The HCP for LT graphs with maximum degree 6 z 1 z 4 z i V(G’) z 2 z 5 Z i is the corresponding Graph G’ z 6 z 3 node in G Z 1 Z 4 Z 2 Graph G Z 5 Z 3 Z 6 Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 11 / 31

  2. The HCP for LT graphs with maximum degree 6 Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 12 / 31

  3. The HCP for LT graphs with maximum degree 6 Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 12 / 31

  4. The HCP for LT graphs with maximum degree 6 Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 12 / 31

  5. The HCP for LT graphs with maximum degree 6 Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 12 / 31

  6. The HCP for LT graphs with maximum degree 6 Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 12 / 31

  7. Locally hamiltonian (LH) graphs Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 13 / 31

  8. Locally hamiltonian (LH) graphs Smallest connected nonhamiltonian LH graph has order 11 and ∆ = 8 (Pareek et al. 1983). Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 13 / 31

  9. Locally hamiltonian (LH) graphs Smallest connected nonhamiltonian LH graph has order 11 and ∆ = 8 (Pareek et al. 1983). Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 13 / 31

  10. Locally hamiltonian (LH) graphs Smallest connected nonhamiltonian LH graph has order 11 and ∆ = 8 (Pareek et al. 1983). HCP NP-complete for ∆ = 9 Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 13 / 31

  11. Hamilton Cycle Problem for LH graphs Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 14 / 31

  12. Hamilton Cycle Problem for LH graphs The HCP for maximally planar graphs is NP-complete (Chv´ atal 1985) Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 14 / 31

  13. Hamilton Cycle Problem for LH graphs The HCP for maximally planar graphs is NP-complete (Chv´ atal 1985) Chv´ atal’s proof is valid for ∆ ≥ 12. Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 14 / 31

  14. Hamilton Cycle Problem for LH graphs The HCP for maximally planar graphs is NP-complete (Chv´ atal 1985) Chv´ atal’s proof is valid for ∆ ≥ 12. Theorem (van Aardt et al. 2016) If G is a connected LH graph with ∆( G ) ≤ 6, then G is hamiltonian. Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 14 / 31

  15. Hamilton Cycle Problem for LH graphs The HCP for maximally planar graphs is NP-complete (Chv´ atal 1985) Chv´ atal’s proof is valid for ∆ ≥ 12. Theorem (van Aardt et al. 2016) If G is a connected LH graph with ∆( G ) ≤ 6, then G is hamiltonian. There exist connected LH graphs with maximum degree 8 that are nonhamiltonian. Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 14 / 31

  16. Locally Hamiltonian Graphs Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 15 / 31

  17. Locally Hamiltonian Graphs Triangle identification G 1 G 2 G v 1 v 2 v u 2 u 1 u w 2 w 1 w Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 15 / 31

  18. Locally Hamiltonian Graphs Triangle identification G 1 G 2 G v 1 v 2 v u 2 u 1 u w 2 w 1 w Theorem Let G 1 and G 2 be two LH graphs, and let G be a graph obtained from G 1 and G 2 by identifying suitable triangles. Then Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 15 / 31

  19. Locally Hamiltonian Graphs Triangle identification G 1 G 2 G v 1 v 2 v u 2 u 1 u w 2 w 1 w Theorem Let G 1 and G 2 be two LH graphs, and let G be a graph obtained from G 1 and G 2 by identifying suitable triangles. Then (a) G is LH . Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 15 / 31

  20. Locally Hamiltonian Graphs Triangle identification G 1 G 2 G v 1 v 2 v u 2 u 1 u w 2 w 1 w Theorem Let G 1 and G 2 be two LH graphs, and let G be a graph obtained from G 1 and G 2 by identifying suitable triangles. Then (a) G is LH . (b) If G 1 and G 2 are planar, then so is G . Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 15 / 31

  21. Locally Hamiltonian Graphs Triangle identification G 1 G 2 G v 1 v 2 v u 2 u 1 u w 2 w 1 w Theorem Let G 1 and G 2 be two LH graphs, and let G be a graph obtained from G 1 and G 2 by identifying suitable triangles. Then (a) G is LH . (b) If G 1 and G 2 are planar, then so is G . (c) If G is hamiltonian, so are both G 1 and G 2 . Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 15 / 31

  22. LH Graphs - the Hamilton Cycle Problem Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 16 / 31

  23. LH Graphs - the Hamilton Cycle Problem Theorem The HCP for LH graphs with ∆ ≥ 9 is NP-complete. Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 16 / 31

  24. LH Graphs - the Hamilton Cycle Problem Theorem The HCP for LH graphs with ∆ ≥ 9 is NP-complete. We will use the same approach as for LT graphs. Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 16 / 31

  25. LH Graphs - the Hamilton Cycle Problem Theorem The HCP for LH graphs with ∆ ≥ 9 is NP-complete. We will use the same approach as for LT graphs. v 1 H D v 2 x 2 v 3 u 3 x 3 x 1 u 2 u 1 (a) (b) The graph H is locally hamiltonian and nonhamiltonian. Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 16 / 31

  26. LH Graphs - the Hamilton Cycle Problem Graph H is combined with two copies of graph D to create the graph F : Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 17 / 31

  27. LH Graphs - the Hamilton Cycle Problem Graph H is combined with two copies of graph D to create the graph F : F i Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 17 / 31

  28. LH Graphs - the Hamilton Cycle Problem Vertices and Nodes and edges in G’ borders in G Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 18 / 31

  29. LH Graphs - the Hamilton Cycle Problem Vertices and Nodes and edges in G’ borders in G Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 18 / 31

  30. LH Graphs - the Hamilton Cycle Problem Vertices and Nodes and edges in G’ borders in G Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 18 / 31

  31. LH Graphs - the Hamilton Cycle Problem z 1 z 4 z i V(G’) z 5 Graph G’ z 2 Z i is the corresponding z 6 z 3 node in G Z 4 Z 1 Graph G Z 2 Z 5 Z 3 Z 6 Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 19 / 31

  32. LH Graphs - the Hamilton Cycle Problem F i Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 20 / 31

  33. LH Graphs - the Hamilton Cycle Problem F i Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 20 / 31

  34. LH Graphs - the Hamilton Cycle Problem F i Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 20 / 31

  35. Locally 2-nested hamiltonian (L2H) graphs Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 21 / 31

  36. Locally 2-nested hamiltonian (L2H) graphs A graph G is L2H if G is LH and � N ( v ) � is LH for any v ∈ V ( G ). Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 21 / 31

  37. Locally 2-nested hamiltonian (L2H) graphs A graph G is L2H if G is LH and � N ( v ) � is LH for any v ∈ V ( G ). Smallest connected nonhamiltonian (L2H) graph has order 13 and ∆ = 10. Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 21 / 31

  38. Locally 2-nested hamiltonian (L2H) graphs A graph G is L2H if G is LH and � N ( v ) � is LH for any v ∈ V ( G ). Smallest connected nonhamiltonian (L2H) graph has order 13 and ∆ = 10. Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 21 / 31

  39. Locally 2-nested hamiltonian (L2H) graphs A graph G is L2H if G is LH and � N ( v ) � is LH for any v ∈ V ( G ). Smallest connected nonhamiltonian (L2H) graph has order 13 and ∆ = 10. HCP NP-complete for ∆ = 13 Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 21 / 31

  40. L2H Graphs - the Hamilton Cycle Problem z 1 z 4 z i V( G’) Graph G’ z 2 z 5 Z i is the corresponding z 6 z 3 node in G Z 1 Z 4 Graph G Z 5 Z 2 Z 6 Z 3 Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 22 / 31

  41. Locally Hamilton-connected graphs Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 23 / 31

  42. Locally Hamilton-connected graphs A graph G is Hamilton-connected if there is a Hamilton path connecting any two vertices u and v in V ( G ). Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 23 / 31

  43. Locally Hamilton-connected graphs A graph G is Hamilton-connected if there is a Hamilton path connecting any two vertices u and v in V ( G ). Smallest connected nonhamiltonian (LHC) graph has order 15 and ∆ = 11. Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 23 / 31

  44. Locally Hamilton-connected graphs A graph G is Hamilton-connected if there is a Hamilton path connecting any two vertices u and v in V ( G ). Smallest connected nonhamiltonian (LHC) graph has order 15 and ∆ = 11. Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 23 / 31

  45. Locally Hamilton-connected graphs A graph G is Hamilton-connected if there is a Hamilton path connecting any two vertices u and v in V ( G ). Smallest connected nonhamiltonian (LHC) graph has order 15 and ∆ = 11. HCP NP-complete for ∆ = 15 Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 23 / 31

  46. LHC Graphs - the Hamilton Cycle Problem The graph G ’ The graph G Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 24 / 31

  47. Locally Chv´ atal-Erd¨ os graphs A graph G is Chv´ atal-Erd¨ os if α ( G ) ≤ κ ( G ). Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 25 / 31

  48. Locally Chv´ atal-Erd¨ os graphs A graph G is Chv´ atal-Erd¨ os if α ( G ) ≤ κ ( G ). A graph is locally Chv´ atal-Erd¨ os if α ( � N ( v ) � ) ≤ κ ( � N ( v ) � ) for any v ∈ V ( G ). Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 25 / 31

  49. Locally Chv´ atal-Erd¨ os graphs A graph G is Chv´ atal-Erd¨ os if α ( G ) ≤ κ ( G ). A graph is locally Chv´ atal-Erd¨ os if α ( � N ( v ) � ) ≤ κ ( � N ( v ) � ) for any v ∈ V ( G ). A graph is closed-locally Chv´ atal-Erd¨ os if α ( � N [ v ] � ) ≤ κ ( � N [ v ] � ) for any v ∈ V ( G ). Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 25 / 31

  50. Locally Chv´ atal-Erd¨ os graphs A graph G is Chv´ atal-Erd¨ os if α ( G ) ≤ κ ( G ). A graph is locally Chv´ atal-Erd¨ os if α ( � N ( v ) � ) ≤ κ ( � N ( v ) � ) for any v ∈ V ( G ). A graph is closed-locally Chv´ atal-Erd¨ os if α ( � N [ v ] � ) ≤ κ ( � N [ v ] � ) for any v ∈ V ( G ). A cl-LCE graph is 1-tough (Chen et al. 2013). Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 25 / 31

  51. Locally Chv´ atal-Erd¨ os graphs A graph G is Chv´ atal-Erd¨ os if α ( G ) ≤ κ ( G ). A graph is locally Chv´ atal-Erd¨ os if α ( � N ( v ) � ) ≤ κ ( � N ( v ) � ) for any v ∈ V ( G ). A graph is closed-locally Chv´ atal-Erd¨ os if α ( � N [ v ] � ) ≤ κ ( � N [ v ] � ) for any v ∈ V ( G ). A cl-LCE graph is 1-tough (Chen et al. 2013). It is not known if cl-LCE graphs are hamiltonian. Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 25 / 31

  52. Discussion Table: The values of key parameters for various local properties. LC LT LH L2H LHC cl-LCE LCE Minimum n ( G ) if G is not 1-tough 5 7 11 13 15 N/A N/A Minimum ∆( G ) if G is not 1-tough 4 5 8 10 11 N/A N/A HCP is NP-complete for ∆( G ) at least 5 6 9* 13* 15* ? ? Minimum degree of local connectedness 1 1 2 3 3 1 2 *It is not known whether these values are best possible. Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 26 / 31

  53. Discussion Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 27 / 31

  54. Discussion We can generalize the concept of L2H graphs to LkH graphs. Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 27 / 31

  55. Discussion We can generalize the concept of L2H graphs to LkH graphs. Locally ( k + 1)-connected Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 27 / 31

  56. Discussion We can generalize the concept of L2H graphs to LkH graphs. Locally ( k + 1)-connected The HCP for LkH graphs is NP-complete for any k ≥ 1. Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 27 / 31

  57. Discussion We can generalize the concept of L2H graphs to LkH graphs. Locally ( k + 1)-connected The HCP for LkH graphs is NP-complete for any k ≥ 1. The important variable is the relationship between the local connectivity and local independence number. Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 27 / 31

  58. Discussion Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 28 / 31

  59. Discussion Oberly-Sumner Conjecture: A connected graph that is locally k -connected and K 1 , k +2 -free is hamiltonian. Johan de Wet et al. (UP, CoE) Local Properties and the HCP Bucharest 2018 28 / 31

Recommend

Rainbow Hamilton cycles Po-Shen Loh Carnegie Mellon University Joint work with Alan Frieze

Rainbow Hamilton cycles Po-Shen Loh Carnegie Mellon University Joint work with Alan Frieze

Rainbow Hamilton cycles Po-Shen Loh Carnegie Mellon University Joint work with Alan Frieze Graphs Definition A cycle is Hamiltonian if it visits every vertex exactly once. Graphs Definition A cycle is Hamiltonian if it visits every vertex

785 views • 54 slides
Even cycle problem for directed graphs Avadhut M. Sardeshmukh Computer Science and Engineering

Even cycle problem for directed graphs Avadhut M. Sardeshmukh Computer Science and Engineering

Introduction Characterization and Lemmas Proof of the theorem References Proofs of Lemmas Even cycle problem for directed graphs Avadhut M. Sardeshmukh Computer Science and Engineering IIT Bombay avadhut@iitb.ac.in Avadhut M. Sardeshmukh

832 views • 42 slides
Graphs whose local graphs are srg with the second eigenvalue 3 A.A. Makhnev Institute of

Graphs whose local graphs are srg with the second eigenvalue 3 A.A. Makhnev Institute of

t1 t2 t3 t4 t4 lit Graphs whose local graphs are srg with the second eigenvalue 3 A.A. Makhnev Institute of Mathematics and Mechanics UB RAS Ekaterinburg, Russia makhnev@imm.uran.ru Villanova, June 2014 A.A. Makhnev Graphs whose local

405 views • 27 slides
Hamilton Decompositions of Infinite Circulant Graphs Sara Herke The University of Queensland

Hamilton Decompositions of Infinite Circulant Graphs Sara Herke The University of Queensland

Hamilton Decompositions of Infinite Circulant Graphs Sara Herke The University of Queensland joint work with Darryn Bryant, Barbara Maenhaut and Bridget Webb January 2017 Infinite Circulant Graphs Infinite Circulant Graphs Group G with

1.25k views • 86 slides
Graph Representation Learning William L. Hamilton COMP 551 Special Topic Lecture Will

Graph Representation Learning William L. Hamilton COMP 551 Special Topic Lecture Will

Graph Representation Learning William L. Hamilton COMP 551 Special Topic Lecture Will Hamilton, McGill and Mila 1 Why graphs? Graphs are a general language for describing and modeling complex systems Will Hamilton, McGill and Mila 2

1.14k views • 74 slides
Line graphs of multigraphs and Hamilton-connectedness of claw-free graphs Zden ek Ryj a

Line graphs of multigraphs and Hamilton-connectedness of claw-free graphs Zden ek Ryj a

Line graphs of multigraphs and Hamilton-connectedness of claw-free graphs Zden ek Ryj a cek, Plze n Joint work with Petr Vr ana, Plze n 1 Graph: no loops, no parallel edges Multigraph: no loops, parallel edges possible c ( G

529 views • 19 slides
Hamilton-Jacobi Skeleton and Shock Graphs Peihong Zhu University of Utah Papers:

Hamilton-Jacobi Skeleton and Shock Graphs Peihong Zhu University of Utah Papers:

Hamilton-Jacobi Skeleton and Shock Graphs Peihong Zhu University of Utah Papers: Hamilton-Jacobi Skeleton (Siddiqi et al.) Shock Grammar (Kimia, Siddiqi) Introduction Skeleton (medial axis) A thin representation of shape. good

168 views • 14 slides
Graphs 3 4 8 1 ORD SFO 802 1743 337 1 2 3 3 LAX DFW Outline / Reading Graphs (6.1)

Graphs 3 4 8 1 ORD SFO 802 1743 337 1 2 3 3 LAX DFW Outline / Reading Graphs (6.1)

Graphs 3 4 8 1 ORD SFO 802 1743 337 1 2 3 3 LAX DFW Outline / Reading Graphs (6.1) Definition Applications Terminology Properties ADT Data structures for graphs (6.2) Edge list structure

443 views • 15 slides
Packing tight Hamilton cycles Po-Shen Loh Carnegie Mellon University Joint work with Alan Frieze

Packing tight Hamilton cycles Po-Shen Loh Carnegie Mellon University Joint work with Alan Frieze

Packing tight Hamilton cycles Po-Shen Loh Carnegie Mellon University Joint work with Alan Frieze and Michael Krivelevich General graphs Definition A cycle is Hamiltonian if it visits every vertex exactly once. (Dirac 52.) Degrees n 2

1.14k views • 74 slides
Hamilton Urban Core Community Health Center Update Hamilton Niagara Haldimand Brant Local Health

Hamilton Urban Core Community Health Center Update Hamilton Niagara Haldimand Brant Local Health

Hamilton Urban Core Community Health Center Update Hamilton Niagara Haldimand Brant Local Health Integration Network Board of Directors August 28, 2013 1 Presentation Outline: Pre-Capital Submission March 2013 Review and

399 views • 19 slides
Denis Corr, Ph.D. Chair Clean Air Hamilton Clean Air Hamilton www.cleanairhamilton.ca Clean

Denis Corr, Ph.D. Chair Clean Air Hamilton Clean Air Hamilton www.cleanairhamilton.ca Clean

Denis Corr, Ph.D. Chair Clean Air Hamilton Clean Air Hamilton www.cleanairhamilton.ca Clean Air Hamilton Members: Local Citizens Ontario MOECC Health Canada Environment Canada ArcelorMittal Dofasco Stelco Alectra Utilities Alectra

441 views • 11 slides
of graphs and pairwise transitive designs CHERYL E PRAEGER CENTRE FOR THE MATHEMATICS OF

of graphs and pairwise transitive designs CHERYL E PRAEGER CENTRE FOR THE MATHEMATICS OF

Local transitivity properties of graphs and pairwise transitive designs CHERYL E PRAEGER CENTRE FOR THE MATHEMATICS OF SYMMETRY AND COMPUTATION CANADAM JUNE, 2013 Interplay between different areas Groups Designs Graphs The University of

297 views • 29 slides
Graphs Outline and Reading Graphs (12.1) Definition Applications Terminology

Graphs Outline and Reading Graphs (12.1) Definition Applications Terminology

1 ORD DFW 802 1233 1843 3 4 7 1 LAX 337 SFO Graphs Graphs Outline and Reading Graphs (12.1) Definition Applications Terminology Properties ADT Data structures for graphs (12.2) Edge list structure

1.34k views • 129 slides
Short cycle covers of graphs with minimum degree three Tom Kaiser, Daniel Krl, Bernard

Short cycle covers of graphs with minimum degree three Tom Kaiser, Daniel Krl, Bernard

Short cycle covers of graphs with minimum degree three Tom Kaiser, Daniel Krl, Bernard Lidick and Pavel Nejedl Department of Applied Math Charles University Cycles and Colourings 2008 - Tatransk trba Short cycle covers of

645 views • 32 slides
Objectives Data structure: Heaps Data structure: Graphs Submit problem set 2 Jan 25, 2019

Objectives Data structure: Heaps Data structure: Graphs Submit problem set 2 Jan 25, 2019

1/25/19 Objectives Data structure: Heaps Data structure: Graphs Submit problem set 2 Jan 25, 2019 CSCI211 - Sprenkle 1 Review What is a priority queue? What is a heap? Properties Implementation What is the process for

215 views • 19 slides
Introduction: Generative Model for Graphs Modeling graphs is fundamental for studying networks

Introduction: Generative Model for Graphs Modeling graphs is fundamental for studying networks

GraphRNN: A Deep Generative Model for Graphs (24 Feb 2018) Jiaxuan You, Rex Ying, Xiang Ren, William L. Hamilton, Jure Leskovec Presented by: Jesse Bettencourt and Harris Chan March 9, 2018 University of Toronto, Vector Institute 1

306 views • 29 slides
Densest/Heaviest k -subgraph on Interval Graphs, Chordal Graphs and Planar Graphs Presented by

Densest/Heaviest k -subgraph on Interval Graphs, Chordal Graphs and Planar Graphs Presented by

Densest/Heaviest k -subgraph on Interval Graphs, Chordal Graphs and Planar Graphs Presented by Jian Li, Fudan University Mar 2007, HKUST May 8, 2006 1 / 25 Problem Definition: Densest k -Subgraph Problem(DS- k ): Input: G ( V, E ) , k > 0 .

398 views • 25 slides
Chromatic properties of Cayley graphs Elena Konstantinova Sobolev Institute of Mathematics

Chromatic properties of Cayley graphs Elena Konstantinova Sobolev Institute of Mathematics

Chromatic properties of Cayley graphs Elena Konstantinova Sobolev Institute of Mathematics Novosibirsk State University Mathematical Colloquium, Ljubljana, Slovenia October 2, 2015 Elena Konstantinova Chromatic properties of Cayley graphs

450 views • 25 slides
Stable Sets in Graphs with Bounded Odd Cycle Packing Number Tony Huynh (Monash) joint with

Stable Sets in Graphs with Bounded Odd Cycle Packing Number Tony Huynh (Monash) joint with

Stable Sets in Graphs with Bounded Odd Cycle Packing Number Tony Huynh (Monash) joint with Michele Conforti, Samuel Fiorini, Gwena el Joret, and Stefan Weltge Maximum Weight Stable Set Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle

794 views • 77 slides
3-coloring triangle-free planar graphs with a precolored 9-cycle ILKYOO CHOI 1 , Jan Ekstein 2 ,

3-coloring triangle-free planar graphs with a precolored 9-cycle ILKYOO CHOI 1 , Jan Ekstein 2 ,

3-coloring triangle-free planar graphs with a precolored 9-cycle 3-coloring triangle-free planar graphs with a precolored 9-cycle ILKYOO CHOI 1 , Jan Ekstein 2 , P remysl Holub 2 , Bernard Lidick y 1 University of Illinois at

1.75k views • 99 slides
What is Mercury? Naturally occurring element (atomic number 80) Heavy metal, can be toxic

What is Mercury? Naturally occurring element (atomic number 80) Heavy metal, can be toxic

The Global Biogeochemical Mercury Cycle What is mercury? Mercury species in nature Their properties and place in the cycle How is mercury converted among its various forms? The mercury cycle: Major steps: in general

533 views • 14 slides
Multi-Cycle CPU: Datapath and Control CSE 141, S2'06 Jeff Brown Why a Multiple Clock Cycle CPU?

Multi-Cycle CPU: Datapath and Control CSE 141, S2'06 Jeff Brown Why a Multiple Clock Cycle CPU?

Multi-Cycle CPU: Datapath and Control CSE 141, S2'06 Jeff Brown Why a Multiple Clock Cycle CPU? the problem => single-cycle cpu has a cycle time long enough to complete the longest instruction in the machine the solution => break

559 views • 38 slides
Outline and Reading Weighted graphs (7.1) Shortest Paths Shortest path problem Shortest

Outline and Reading Weighted graphs (7.1) Shortest Paths Shortest path problem Shortest

Outline and Reading Weighted graphs (7.1) Shortest Paths Shortest path problem Shortest path properties Dijkstras algorithm (7.1.1) 0 A 4 8 Algorithm 2 8 2 3 Edge relaxation 7 1 B C D The Bellman-Ford

348 views • 4 slides
Sequence Alignment (chapter 6) The biological problem l Global alignment l Local alignment l

Sequence Alignment (chapter 6) The biological problem l Global alignment l Local alignment l

Sequence Alignment (chapter 6) The biological problem l Global alignment l Local alignment l Multiple alignment l Introduction to bioinformatics, Autumn 2006 22 Background: comparative genomics Basic question in biology: what properties

934 views • 44 slides

More recommend