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Computational Combinatorics and the search for Uniquely K r -Saturated Graphs Stephen G. Hartke University of NebraskaLincoln, USA hartke@math.unl.edu http://www.math.unl.edu/ shartke2/ Joint work with Derrick Stolee, Iowa State

  1. Multiple paths! Unlabeled graphs

  2. Multiple paths! Unlabeled graphs

  3. Multiple paths! Unlabeled graphs

  4. Goal: Exactly one path. Unlabeled graphs

  5. Goal: Exactly one path. Unlabeled graphs

  6. Partition by subtrees. Unlabeled graphs

  7. Parallelize!

  8. Overview Search as a Poset Implementation The TreeSearch library enables parallelization in the Condor scheduler. Executes on the Open Science Grid , a collection of supercomputers around the country. Stephen Hartke (UNL) Uniquely K r -Saturated Graphs 17 / 1

  9. Overview Search as a Poset Computational Combinatorics TreeSearch High Performance Algorithms Computing Computational Combinatorics Pure Combinatorics Stephen Hartke (UNL) Uniquely K r -Saturated Graphs 18 / 1

  10. Uniquely K r -Saturated Graphs H -Saturated Graphs H -Saturated Graphs Definition A graph G is H -saturated if ◦ G does not contain H as a subgraph. ( H -free ) ◦ For every e ∈ E ( G ) , G + e contains H as a subgraph. 5-cycle 6-cycle Example: H = K 3 where K r is the complete graph on r vertices. Stephen Hartke (UNL) Uniquely K r -Saturated Graphs 19 / 1

  11. Uniquely K r -Saturated Graphs H -Saturated Graphs H -Saturated Graphs Definition A graph G is H -saturated if ◦ G does not contain H as a subgraph. ( H -free ) ◦ For every e ∈ E ( G ) , G + e contains H as a subgraph. 5-cycle 6-cycle is K 3 -saturated Example: H = K 3 where K r is the complete graph on r vertices. Stephen Hartke (UNL) Uniquely K r -Saturated Graphs 19 / 1

  12. Uniquely K r -Saturated Graphs H -Saturated Graphs H -Saturated Graphs Definition A graph G is H -saturated if ◦ G does not contain H as a subgraph. ( H -free ) ◦ For every e ∈ E ( G ) , G + e contains H as a subgraph. 5-cycle 6-cycle is K 3 -saturated Example: H = K 3 where K r is the complete graph on r vertices. Stephen Hartke (UNL) Uniquely K r -Saturated Graphs 19 / 1

  13. Uniquely K r -Saturated Graphs H -Saturated Graphs H -Saturated Graphs Definition A graph G is H -saturated if ◦ G does not contain H as a subgraph. ( H -free ) ◦ For every e ∈ E ( G ) , G + e contains H as a subgraph. 5-cycle 6-cycle is K 3 -saturated is not K 3 -saturated Example: H = K 3 where K r is the complete graph on r vertices. Stephen Hartke (UNL) Uniquely K r -Saturated Graphs 19 / 1

  14. Uniquely K r -Saturated Graphs H -Saturated Graphs Tur´ an’s Theorem Theorem (Tur´ an, 1941) Let r ≥ 3. If G is K r -saturated on n vertices, � n 2 1 � then G has at most 1 − 2 edges (asymptotically). r − 1 Stephen Hartke (UNL) Uniquely K r -Saturated Graphs 20 / 1

  15. Uniquely K r -Saturated Graphs H -Saturated Graphs Tur´ an’s Theorem Theorem (Tur´ an, 1941) Let r ≥ 3. If G is K r -saturated on n vertices, � n 2 1 � then G has at most 1 − 2 edges (asymptotically). r − 1 r − 1 parts Stephen Hartke (UNL) Uniquely K r -Saturated Graphs 20 / 1

  16. Uniquely K r -Saturated Graphs H -Saturated Graphs Tur´ an’s Theorem Theorem (Tur´ an, 1941) Let r ≥ 3. If G is K r -saturated on n vertices, � n 2 1 � then G has at most 1 − 2 edges (asymptotically). r − 1 r − 1 parts Many copies of K r ! Stephen Hartke (UNL) Uniquely K r -Saturated Graphs 20 / 1

  17. Uniquely K r -Saturated Graphs H -Saturated Graphs Erd˝ os, Hajnal, and Moon Theorem (Erd˝ os, Hajnal, Moon, 1964) Let r ≥ 3. If G is K r -saturated on n vertices, then G has at least ( r − 2 2 ) + ( r − 2 )( n − r + 2 ) edges. Stephen Hartke (UNL) Uniquely K r -Saturated Graphs 21 / 1

  18. Uniquely K r -Saturated Graphs H -Saturated Graphs Erd˝ os, Hajnal, and Moon Theorem (Erd˝ os, Hajnal, Moon, 1964) Let r ≥ 3. If G is K r -saturated on n vertices, then G has at least ( r − 2 2 ) + ( r − 2 )( n − r + 2 ) edges. 1-book 2-book 3-book Stephen Hartke (UNL) Uniquely K r -Saturated Graphs 21 / 1

  19. Uniquely K r -Saturated Graphs H -Saturated Graphs Erd˝ os, Hajnal, and Moon Theorem (Erd˝ os, Hajnal, Moon, 1964) Let r ≥ 3. If G is K r -saturated on n vertices, then G has at least ( r − 2 2 ) + ( r − 2 )( n − r + 2 ) edges. 1-book 2-book 3-book Exactly one copy of K r ! Stephen Hartke (UNL) Uniquely K r -Saturated Graphs 21 / 1

  20. Uniquely H -Saturated Graphs Definition Uniquely H -Saturated Graphs The Tur´ an graph has many copies of K r when an edge is added. The books have exactly one copy of K r when an edge is added. Stephen Hartke (UNL) Uniquely K r -Saturated Graphs 22 / 1

  21. Uniquely H -Saturated Graphs Definition Uniquely H -Saturated Graphs The Tur´ an graph has many copies of K r when an edge is added. The books have exactly one copy of K r when an edge is added. Definition A graph G is uniquely H -saturated if G does not contain H as a subgraph and for every edge e ∈ G admits exactly one copy of H in G + e . Stephen Hartke (UNL) Uniquely K r -Saturated Graphs 22 / 1

  22. Uniquely H -Saturated Graphs Uniquely C k -Saturated Graphs Uniquely C k -Saturated Graphs Lemma (Cooper, Lenz, LeSaulnier, Wenger, West, 2011) The uniquely C 3 -saturated graphs are either stars or Moore graphs of diameter 2 and girth 5. Stephen Hartke (UNL) Uniquely K r -Saturated Graphs 23 / 1

  23. Uniquely H -Saturated Graphs Uniquely C k -Saturated Graphs Uniquely C k -Saturated Graphs Lemma (Cooper, Lenz, LeSaulnier, Wenger, West, 2011) The uniquely C 3 -saturated graphs are either stars or Moore graphs of diameter 2 and girth 5. Theorem (Hoffman, Singleton, 1964) There are a finite number of Moore graphs of diameter 2 and girth 5. Stephen Hartke (UNL) Uniquely K r -Saturated Graphs 23 / 1

  24. Uniquely H -Saturated Graphs Uniquely C k -Saturated Graphs Uniquely C k -Saturated Graphs Lemma (Cooper, Lenz, LeSaulnier, Wenger, West, 2011) The uniquely C 3 -saturated graphs are either stars or Moore graphs of diameter 2 and girth 5. Theorem (Hoffman, Singleton, 1964) There are a finite number of Moore graphs of diameter 2 and girth 5. ? C 5 Petersen Hoffman– 57-Regular Singleton Order 3250 Stephen Hartke (UNL) Uniquely K r -Saturated Graphs 23 / 1

  25. Uniquely H -Saturated Graphs Uniquely C k -Saturated Graphs Uniquely C k -Saturated Graphs Theorem (Cooper, Lenz, LeSaulnier, Wenger, West, 2011) There are a finite number of uniquely C 4 -saturated graphs. Stephen Hartke (UNL) Uniquely K r -Saturated Graphs 24 / 1

  26. Uniquely H -Saturated Graphs Uniquely C k -Saturated Graphs Uniquely C k -Saturated Graphs Theorem (Cooper, Lenz, LeSaulnier, Wenger, West, 2011) There are a finite number of uniquely C 4 -saturated graphs. Theorem (Wenger, 2010) The only uniquely C 5 -saturated graphs are friendship graphs . Stephen Hartke (UNL) Uniquely K r -Saturated Graphs 24 / 1

  27. Uniquely H -Saturated Graphs Uniquely C k -Saturated Graphs Uniquely C k -Saturated Graphs Theorem (Cooper, Lenz, LeSaulnier, Wenger, West, 2011) There are a finite number of uniquely C 4 -saturated graphs. Theorem (Wenger, 2010) The only uniquely C 5 -saturated graphs are friendship graphs . Theorem (Wenger, 2010) For k ∈ { 6 , 7 , 8 } , no uniquely C k -saturated graph exists. Conjecture (Wenger, 2010) For k ≥ 9, no uniquely C k -saturated graph exists. Stephen Hartke (UNL) Uniquely K r -Saturated Graphs 24 / 1

  28. Uniquely H -Saturated Graphs Definition Uniquely K r -Saturated Graphs We consider the case where H = K r (an r -clique ) for r ≥ 4. Stephen Hartke (UNL) Uniquely K r -Saturated Graphs 25 / 1

  29. Uniquely H -Saturated Graphs Definition Uniquely K r -Saturated Graphs We consider the case where H = K r (an r -clique ) for r ≥ 4. ( K 3 ∼ = C 3 ) Stephen Hartke (UNL) Uniquely K r -Saturated Graphs 25 / 1

  30. Uniquely H -Saturated Graphs Definition Uniquely K r -Saturated Graphs We consider the case where H = K r (an r -clique ) for r ≥ 4. ( K 3 ∼ = C 3 ) Definition A graph G is uniquely K r -saturated if G does not contain an r -clique and for every edge e ∈ G there is exactly one r -clique in G + e . Stephen Hartke (UNL) Uniquely K r -Saturated Graphs 25 / 1

  31. Uniquely H -Saturated Graphs Definition Dominating Vertices Adding a dominating vertex to a uniquely K r -saturated graph creates a uniquely K r + 1 -saturated graph. Stephen Hartke (UNL) Uniquely K r -Saturated Graphs 26 / 1

  32. Uniquely H -Saturated Graphs Definition Dominating Vertices Adding a dominating vertex to a uniquely K r -saturated graph creates a uniquely K r + 1 -saturated graph. Stephen Hartke (UNL) Uniquely K r -Saturated Graphs 26 / 1

  33. Uniquely H -Saturated Graphs Definition Dominating Vertices Adding a dominating vertex to a uniquely K r -saturated graph creates a uniquely K r + 1 -saturated graph. Stephen Hartke (UNL) Uniquely K r -Saturated Graphs 26 / 1

  34. Uniquely H -Saturated Graphs Definition Dominating Vertices Adding a dominating vertex to a uniquely K r -saturated graph creates a uniquely K r + 1 -saturated graph. Stephen Hartke (UNL) Uniquely K r -Saturated Graphs 26 / 1

  35. Uniquely H -Saturated Graphs Definition Dominating Vertices Adding a dominating vertex to a uniquely K r -saturated graph creates a uniquely K r + 1 -saturated graph. Call uniquely K r -saturated graphs without a dominating vertex r -primitive . Stephen Hartke (UNL) Uniquely K r -Saturated Graphs 26 / 1

  36. Uniquely H -Saturated Graphs Known r -Primitive Graphs r -Primitive Graphs A uniquely K r -saturated graph with no dominating vertex is r -primitive . Stephen Hartke (UNL) Uniquely K r -Saturated Graphs 27 / 1

  37. Uniquely H -Saturated Graphs Known r -Primitive Graphs r -Primitive Graphs A uniquely K r -saturated graph with no dominating vertex is r -primitive . 2 -primitive graphs are empty graphs . Stephen Hartke (UNL) Uniquely K r -Saturated Graphs 27 / 1

  38. Uniquely H -Saturated Graphs Known r -Primitive Graphs r -Primitive Graphs A uniquely K r -saturated graph with no dominating vertex is r -primitive . 2 -primitive graphs are empty graphs . 3 -primitive graphs are Moore graphs of diameter 2 and girth 5. ? C 5 Petersen Hoffman– 57-Regular Singleton Order 3250 Stephen Hartke (UNL) Uniquely K r -Saturated Graphs 27 / 1

  39. Uniquely H -Saturated Graphs Known r -Primitive Graphs r -Primitive Graphs A uniquely K r -saturated graph with no dominating vertex is r -primitive . For r ≥ 1, C 2 r − 1 is r -primitive. C 5 C 7 C 9 (Collins, Cooper, Kay, Wenger, 2010) Stephen Hartke (UNL) Uniquely K r -Saturated Graphs 28 / 1

  40. Uniquely H -Saturated Graphs Known r -Primitive Graphs r -Primitive Graphs A uniquely K r -saturated graph with no dominating vertex is r -primitive . For r ≥ 1, C 2 r − 1 is r -primitive. C 5 C 7 C 9 (Collins, Cooper, Kay, Wenger, 2010) Stephen Hartke (UNL) Uniquely K r -Saturated Graphs 28 / 1

  41. Uniquely H -Saturated Graphs Known r -Primitive Graphs r -Primitive Graphs A uniquely K r -saturated graph with no dominating vertex is r -primitive . For r ≥ 1, C 2 r − 1 is r -primitive. C 5 C 7 C 9 (Collins, Cooper, Kay, Wenger, 2010) Stephen Hartke (UNL) Uniquely K r -Saturated Graphs 28 / 1

  42. Uniquely H -Saturated Graphs Known r -Primitive Graphs r -Primitive Graphs A uniquely K r -saturated graph with no dominating vertex is r -primitive . For r ≥ 1, C 2 r − 1 is r -primitive. C 5 C 7 C 9 (Collins, Cooper, Kay, Wenger, 2010) Stephen Hartke (UNL) Uniquely K r -Saturated Graphs 28 / 1

  43. Uniquely H -Saturated Graphs Known r -Primitive Graphs Uniquely K 4 -Saturated Graphs 10 vertices 12 vertices Previously known 4-primitive graphs (Collins, Cooper, Kay, 2010) Stephen Hartke (UNL) Uniquely K r -Saturated Graphs 29 / 1

  44. Uniquely H -Saturated Graphs Known r -Primitive Graphs Computational Combinatorics TreeSearch High Performance Algorithms Computing Computational Combinatorics Pure Problem Combinatorics Stephen Hartke (UNL) Uniquely K r -Saturated Graphs 30 / 1

  45. Uniquely H -Saturated Graphs Main Questions The Problem Goal: Characterize uniquely K r -saturated graphs. First Step: Reduce to r -primitive graphs. Stephen Hartke (UNL) Uniquely K r -Saturated Graphs 31 / 1

  46. Uniquely H -Saturated Graphs Main Questions The Problem Goal: Characterize uniquely K r -saturated graphs. First Step: Reduce to r -primitive graphs. 1. Fix r ≥ 3. Are there a finite number of r -primitive graphs? Stephen Hartke (UNL) Uniquely K r -Saturated Graphs 31 / 1

  47. Uniquely H -Saturated Graphs Main Questions The Problem Goal: Characterize uniquely K r -saturated graphs. First Step: Reduce to r -primitive graphs. 1. Fix r ≥ 3. Are there a finite number of r -primitive graphs? 2. Is every r -primitive graph regular ? Stephen Hartke (UNL) Uniquely K r -Saturated Graphs 31 / 1

  48. Uniquely H -Saturated Graphs Computational Method Edges and Non-Edges Non-edges are crucial to the structure of r -primitive graphs. Stephen Hartke (UNL) Uniquely K r -Saturated Graphs 32 / 1

  49. Uniquely H -Saturated Graphs Computational Method Edges and Non-Edges Non-edges are crucial to the structure of r -primitive graphs. Unassigned Edge Non-edge Tricolored graph Stephen Hartke (UNL) Uniquely K r -Saturated Graphs 32 / 1

  50. Uniquely H -Saturated Graphs Computational Method Edges, Non-Edges, and Variables Fix a vertex set { v 1 , v 2 , . . . , v n } . For i , j ∈ { 1 , . . . , n } , let  v i v j ∈ E ( G ) 1   x i , j = 0 v i v j / ∈ E ( G ) .  ∗ v i v j unassigned  A vector x = ( x i , j : i , j ∈ { 1 , . . . , n } ) is a variable assignment . Stephen Hartke (UNL) Uniquely K r -Saturated Graphs 33 / 1

  51. Uniquely H -Saturated Graphs Computational Method Symmetries of the System The constraints ◦ There is no r -clique in G . ◦ Every non-edge e of G has exactly one r -clique in G + e . are independent of vertex labels. Stephen Hartke (UNL) Uniquely K r -Saturated Graphs 34 / 1

  52. Uniquely H -Saturated Graphs Computational Method Symmetries of the System The constraints ◦ There is no r -clique in G . ◦ Every non-edge e of G has exactly one r -clique in G + e . are independent of vertex labels. Automorphisms of the tricolored graph define orbits on variables x i , j . Stephen Hartke (UNL) Uniquely K r -Saturated Graphs 34 / 1

  53. Uniquely H -Saturated Graphs Orbital Branching Orbital Branching Orbital branching reduces the number of isomorphic duplicates. (Ostrowski, Linderoth, Rossi, Smriglio, 2007) Generalizes branch-and-bound strategy from Integer Programming. Stephen Hartke (UNL) Uniquely K r -Saturated Graphs 35 / 1

  54. Uniquely H -Saturated Graphs Orbital Branching Branch-and-Bound x is given Variable x i , j is selected Stephen Hartke (UNL) Uniquely K r -Saturated Graphs 36 / 1

  55. Uniquely H -Saturated Graphs Orbital Branching Branch-and-Bound x is given Variable x i , j is selected x i , j = 0 Stephen Hartke (UNL) Uniquely K r -Saturated Graphs 36 / 1

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