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Algorithmic Graph Minors: Turning Combinatorics to Algorithms Page 1/1111 Antichain: an Infinite Sequence on Non- ≤-Comparable Elements Graph Minors and algorithms The irrelevant vertex technique Algorithmic Graph Minors: turning Combinatorics to Algorithms Dimitrios M. Thilikos Department of Mathematics { µQλ8 UoA - National and Kapodistrian University of Athens Laboratoire d'Informatique, de Robotique et de Micro´electroniquede Montpellier (LIRMM) Montpellier, February 2, 2012 Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 1/1111 Antichain: an infinite sequence on non- ≤-comparable elements. We say that X is Well-Quasi-Ordered by ≤ if it has no infinite antichain Examples: I 2N is not W.Q.O. by set inclusion. I N is not W.Q.O. by divisibility. k I N is W.Q.O. by by component-wise ordering. Graph Minors and algorithms The irrelevant vertex technique Well Quasi Ordering Theory Let X be a set and let \≤" be a partial ordering relation on X . Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 2/1112 We say that X is Well-Quasi-Ordered by ≤ if it has no infinite antichain Examples: I 2N is not W.Q.O. by set inclusion. I N is not W.Q.O. by divisibility. k I N is W.Q.O. by by component-wise ordering. Graph Minors and algorithms The irrelevant vertex technique Well Quasi Ordering Theory Let X be a set and let \≤" be a partial ordering relation on X . Antichain: an infinite sequence on non- ≤-comparable elements. Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 3/1112 Examples: I 2N is not W.Q.O. by set inclusion. I N is not W.Q.O. by divisibility. k I N is W.Q.O. by by component-wise ordering. Graph Minors and algorithms The irrelevant vertex technique Well Quasi Ordering Theory Let X be a set and let \≤" be a partial ordering relation on X . Antichain: an infinite sequence on non- ≤-comparable elements. We say that X is Well-Quasi-Ordered by ≤ if it has no infinite antichain Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 4/1112 Graph Minors and algorithms The irrelevant vertex technique Well Quasi Ordering Theory Let X be a set and let \≤" be a partial ordering relation on X . Antichain: an infinite sequence on non- ≤-comparable elements. We say that X is Well-Quasi-Ordered by ≤ if it has no infinite antichain Examples: I 2N is not W.Q.O. by set inclusion. I N is not W.Q.O. by divisibility. k I N is W.Q.O. by by component-wise ordering. Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 5/1112 The theory of Well-quasi-ordering was first developed by Graham Higman and Erd}os& Rado under the name \finite basis property" Remind: This talk is about graphs and algorithms! Graph Minors and algorithms The irrelevant vertex technique Well Quasi Ordering Theory General question: Given a set X and an ordering relation ≤ on it, is X W.Q.O. by to ≤? Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 6/1113 Remind: This talk is about graphs and algorithms! Graph Minors and algorithms The irrelevant vertex technique Well Quasi Ordering Theory General question: Given a set X and an ordering relation ≤ on it, is X W.Q.O. by to ≤? The theory of Well-quasi-ordering was first developed by Graham Higman and Erd}os& Rado under the name \finite basis property" Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 7/1113 Graph Minors and algorithms The irrelevant vertex technique Well Quasi Ordering Theory General question: Given a set X and an ordering relation ≤ on it, is X W.Q.O. by to ≤? The theory of Well-quasi-ordering was first developed by Graham Higman and Erd}os& Rado under the name \finite basis property" Remind: This talk is about graphs and algorithms! Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 8/1113 v 1 nv: vertex removal: e 2 ne: edge removal: e 3 =e: edge contraction Minor Relation: H ≤ G if H can be obtained from G after a sequence of the above operations Graph Minors and algorithms The irrelevant vertex technique The minor relation on Graphs Graph Minors We define3 local operations on graphs: Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 9/1114 e 2 ne: edge removal: e 3 =e: edge contraction Minor Relation: H ≤ G if H can be obtained from G after a sequence of the above operations Graph Minors and algorithms The irrelevant vertex technique The minor relation on Graphs Graph Minors We define3 local operations on graphs: v 1 nv: vertex removal: Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 10/1114 e 3 =e: edge contraction Minor Relation: H ≤ G if H can be obtained from G after a sequence of the above operations Graph Minors and algorithms The irrelevant vertex technique The minor relation on Graphs Graph Minors We define3 local operations on graphs: v 1 nv: vertex removal: e 2 ne: edge removal: Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 11/1114 Minor Relation: H ≤ G if H can be obtained from G after a sequence of the above operations Graph Minors and algorithms The irrelevant vertex technique The minor relation on Graphs Graph Minors We define3 local operations on graphs: v 1 nv: vertex removal: e 2 ne: edge removal: e 3 =e: edge contraction Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 12/1114 Graph Minors and algorithms The irrelevant vertex technique The minor relation on Graphs Graph Minors We define3 local operations on graphs: v 1 nv: vertex removal: e 2 ne: edge removal: e 3 =e: edge contraction Minor Relation: H ≤ G if H can be obtained from G after a sequence of the above operations Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 13/1114 Graph Minors and algorithms The irrelevant vertex technique Wagner's Conjecture Wagner's Conjecture: I The set of all graphs is W.Q.O. by the minor relation [formulated by Klaus Wagner in the 1930s (?)] Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 14/1115 Graph Minors and algorithms The irrelevant vertex technique Wagner's Conjecture Wagner's Conjecture: I The set of all graphs is W.Q.O. by the minor relation [formulated by Klaus Wagner in the 1930s (?)] Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 15/1115 Now it is known as the Robertson & Seymour Theorem. Width of the proof: < 10 cm (23 papers) 10/11 Fulkerson Prize (2006) (2/3 for N.R. & 4/5 for P.S.) Graph Minors and algorithms The irrelevant vertex technique The Graph Minors Series This conjecture was proven by Neil Robertson and Paul Seymour in their Graph Minor series of papers. Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 16/1116 Now it is known as the Robertson & Seymour Theorem. Width of the proof: < 10 cm (23 papers) 10/11 Fulkerson Prize (2006) (2/3 for N.R. & 4/5 for P.S.) Graph Minors and algorithms The irrelevant vertex technique The Graph Minors Series This conjecture was proven by Neil Robertson and Paul Seymour in their Graph Minor series of papers. Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 17/1116 Now it is known as the Robertson & Seymour Theorem. Width of the proof: < 10 cm (23 papers) 10/11 Fulkerson Prize (2006) (2/3 for N.R. & 4/5 for P.S.) Graph Minors and algorithms The irrelevant vertex technique The Graph Minors Series This conjecture was proven by Neil Robertson and Paul Seymour in their Graph Minor series of papers. Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 18/1116 Width of the proof: < 10 cm (23 papers) 10/11 Fulkerson Prize (2006) (2/3 for N.R. & 4/5 for P.S.) Graph Minors and algorithms The irrelevant vertex technique The Graph Minors Series This conjecture was proven by Neil Robertson and Paul Seymour in their Graph Minor series of papers. Now it is known as the Robertson & Seymour Theorem. Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 19/1116 (23 papers) 10/11 Fulkerson Prize (2006) (2/3 for N.R. & 4/5 for P.S.) Graph Minors and algorithms The irrelevant vertex technique The Graph Minors Series This conjecture was proven by Neil Robertson and Paul Seymour in their Graph Minor series of papers. Now it is known as the Robertson & Seymour Theorem. Width of the proof: < 10 cm Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 20/1116 10/11 Fulkerson Prize (2006) (2/3 for N.R. & 4/5 for P.S.) Graph Minors and algorithms The irrelevant vertex technique The Graph Minors Series This conjecture was proven by Neil Robertson and Paul Seymour in their Graph Minor series of papers. Now it is known as the Robertson & Seymour Theorem. Width of the proof: < 10 cm (23 papers) Dimitrios M. Thilikos Department of Mathematics, UoA Algorithmic Graph Minors: turning Combinatorics to Algorithms Page 21/1116 Graph Minors and algorithms The irrelevant vertex technique The Graph Minors Series This conjecture was proven by Neil Robertson and Paul Seymour in their Graph Minor series of papers.
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