Tracking down graphs that revolt against upper bounds on the - PowerPoint PPT Presentation

The Combinatorial Optimization Workshop 04.-08. January 2016 Aussois Tracking down graphs that revolt against upper bounds on the maximum degree Vera Weil Chair of Management Science Group Schrader School of Business and Economics Department of Computer Science RWTH Aachen University University of Cologne

Introduction G Vera Weil 2 / 11

Introduction G Vera Weil 2 / 11

Introduction G maximum clique size: ω ( G ) Vera Weil 2 / 11

Introduction G maximum clique size: ω ( G ) Vera Weil 2 / 11

Introduction G degree: at least ω ( G ) − 1 maximum clique size: ω ( G ) Vera Weil 2 / 11

Introduction G degree: at least ω ( G ) − 1 maximum clique ⇒ ω ( G ) − 1 ≤ ∆( G ) size: ω ( G ) Vera Weil 2 / 11

Introduction G degree: at least ω ( G ) − 1 maximum clique ⇒ ω ( G ) − 1 ≤ ∆( G ) size: ω ( G ) ω ≤ ∆ + 1 Vera Weil 2 / 11

Introduction ω ≤ ∆ + 1 ⊙ ⊙ � Vera Weil 3 / 11

Introduction ω ≤ ∆ + 1 ∆ + 1 ≤ ω ⊙ ⊙ � Vera Weil 3 / 11

Introduction ω ≤ ∆ + 1 ∆ + 1 ≤ ω ⊙ ⊙ ⊙ ⊙ � � Vera Weil 3 / 11

Introduction ω ≤ ∆ + 1 ∆ + 1 ≤ ω ⊙ ⊙ ⊙ ⊙ � � ∆ = 2, ω = 2 Vera Weil 3 / 11

Introduction ω ≤ ∆ + 1 ∆ + 1 ≤ ω ⊙ ⊙ ⊙ ⊙ � � ∆ = 2, ω = 2 What about ∆ + 1 ≤ ω + 1 ? Vera Weil 3 / 11

Introduction ω ≤ ∆ + 1 ∆ + 1 ≤ ω ⊙ ⊙ ⊙ ⊙ � � ∆ = 2, ω = 2 What about ∆ + 1 ≤ ω + 1 ? Vera Weil 3 / 11

Introduction ω ≤ ∆ + 1 ∆ + 1 ≤ ω ⊙ ⊙ ⊙ ⊙ � � ∆ = 2, ω = 2 What about ∆ + 1 ≤ ω + 1 ? ∆ = 3, ω = 2 Vera Weil 3 / 11

Introduction ω ≤ ∆ + 1 ∆ + 1 ≤ ω ⊙ ⊙ ⊙ ⊙ � ∆ = 2, ω = 2 What about ∆ + 1 ≤ ω + 1 ? ∆ = 3, ω = 2 . . . Vera Weil 3 / 11

Introduction Can we find a function f such that for all graphs G , ∆( G ) ≤ f ( ω ( G )) holds? Vera Weil 4 / 11

Introduction Can we find a function f such that for all graphs G , ∆( G ) ≤ f ( ω ( G )) holds? No! Vera Weil 4 / 11

Introduction Can we find a function f such that for all graphs G , ∆( G ) ≤ f ( ω ( G )) holds? No! For every function f with ∆( G ) ≤ f ( ω ( G )) we find this opponent: Vera Weil 4 / 11

Introduction Can we find a function f such that for all graphs G , ∆( G ) ≤ f ( ω ( G )) holds? No! For every function f with K 1 , f (2)+1 ∆( G ) ≤ f ( ω ( G )) . . . ∆( G ) > f ( ω ( G )) we find this opponent: f (2) + 1 f (2) Vera Weil 4 / 11

Introduction For all functions f we find an infinite number of rebels G that deny ∆( G ) ≤ f ( ω ( G )). Vera Weil 5 / 11

Introduction For all functions f we find an infinite number of rebels G that deny ∆( G ) ≤ f ( ω ( G )). For example: K 1 , f (2)+1 K 1 , f (2)+2 K 1 , f (2)+3 . . . Vera Weil 5 / 11

Introduction For all functions f we find an infinite number of rebels G that deny ∆( G ) ≤ f ( ω ( G )). For example: K 1 , f (2)+1 ← is an inclusionwise minimal rebel! K 1 , f (2)+2 K 1 , f (2)+3 . . . Vera Weil 5 / 11

Introduction For all functions f we find an infinite number of rebels G that deny ∆( G ) ≤ f ( ω ( G )). For example: K 1 , f (2)+1 ← is an inclusionwise minimal rebel! K 1 , f (2)+2 K 1 , f (2)+3 . . . Is the number of inclusionwise minimal rebels finite ? Vera Weil 5 / 11

Is the number of inclusionwise minimal rebels finite ? Vera Weil 6 / 11

Is the number of inclusionwise minimal rebels finite ? Example: Vera Weil 7 / 11

Is the number of inclusionwise minimal rebels finite ? Example: Answer is YES for f ( ω ) = ω + k − 1, for all k ∈ N 0 . (Schaudt & Weil 2015) Vera Weil 7 / 11

Is the number of inclusionwise minimal rebels finite ? Example: Answer is YES for f ( ω ) = ω + k − 1, for all k ∈ N 0 . (Schaudt & Weil 2015) ⇓ Polynomial time algorithm to recognize graphs G where ∆( H ) ≤ ω ( H ) + k − 1 for every H ⊆ G Vera Weil 7 / 11

Is the number of inclusionwise minimal rebels finite ? Example: Answer is YES for f ( ω ) = ω + k − 1, for all k ∈ N 0 . (Schaudt & Weil 2015) ⇓ Polynomial time algorithm to recognize graphs G where ∆( H ) ≤ ω ( H ) + k − 1 for every H ⊆ G ⇓ for some special multigraphs, the chromatic index can be computed in polynomial time Vera Weil 7 / 11

Is the number of inclusionwise minimal rebels finite ? Let G be a hereditary graph class. For every monotone function f s.t. ∆( G ) ≤ f ( ω ( G )) is a rule in G we find p ∈ N such that K 1 , p is a rebel of G . Vera Weil 8 / 11

Is the number of inclusionwise minimal rebels finite ? Let G be a hereditary graph class. For every monotone function f s.t. ∆( G ) ≤ f ( ω ( G )) is a rule in G we find p ∈ N such that K 1 , p is a rebel of G . But also the reverse is true: K 1 , p -rebel generates a rule for the whole graph class! Vera Weil 8 / 11

Is the number of inclusionwise minimal rebels finite ? Let G be a hereditary graph class. For every monotone function f s.t. ∆( G ) ≤ f ( ω ( G )) is a rule in G we find p ∈ N such that K 1 , p is a rebel of G . But also the reverse is true: K 1 , p -rebel generates a rule for the whole graph class! Let G be a hereditary graph class. For every p ∈ N such that K 1 , p is a rebel of G we find a monotone function f s.t. ∆( G ) ≤ f ( ω ( G )) is a rule in G . Vera Weil 8 / 11

Is the number of inclusionwise minimal rebels finite ? Let G be a hereditary graph class. For every p ∈ N such that K 1 , p is a rebel of G we find a monotone function f s.t. ∆( G ) ≤ f ( ω ( G )) is a rule in G . Vera Weil 8 / 11

Is the number of inclusionwise minimal rebels finite ? Let G be a hereditary graph class. For every p ∈ N such that K 1 , p is a rebel of G we find a monotone function f s.t. ∆( G ) ≤ f ( ω ( G )) is a rule in G . Vera Weil 9 / 11

Is the number of inclusionwise minimal rebels finite ? Let G be a hereditary graph class. For every p ∈ N such that K 1 , p is a rebel of G we find a monotone function f s.t. ∆( G ) ≤ f ( ω ( G )) is a rule in G . f ( k ) = sup { ∆( G ) : G ∈ G , ω ( G ) ≤ k } Vera Weil 9 / 11

Is the number of inclusionwise minimal rebels finite ? Let G be a hereditary graph class. For every p ∈ N such that K 1 , p is a rebel of G we find a monotone function f s.t. ∆( G ) ≤ f ( ω ( G )) is a rule in G . f ( k ) = sup { ∆( G ) : G ∈ G , ω ( G ) ≤ k } < R ( k − 1 , p − 1) Vera Weil 9 / 11

Is the number of inclusionwise minimal rebels finite ? Let G be a hereditary graph class. For every p ∈ N such that K 1 , p is a rebel of G we find a monotone function f s.t. ∆( G ) ≤ f ( ω ( G )) is a rule in G . f ( k ) = sup { ∆( G ) : G ∈ G , ω ( G ) ≤ k } < R ( k − 1 , p − 1) ω ≥ k or α ≥ p Vera Weil 9 / 11

Is the number of inclusionwise minimal rebels finite ? Let G be a hereditary graph class. For every p ∈ N such that K 1 , p is a rebel of G we find a monotone function f s.t. ∆( G ) ≤ f ( ω ( G )) is a rule in G . f ( k ) = sup { ∆( G ) : G ∈ G , ω ( G ) ≤ k } < R ( k − 1 , p − 1) ω ≥ k or α ≥ p G ∈ G with ω ( G ) ≤ k Vera Weil 9 / 11

Is the number of inclusionwise minimal rebels finite ? Let G be a hereditary graph class. For every p ∈ N such that K 1 , p is a rebel of G we find a monotone function f s.t. ∆( G ) ≤ f ( ω ( G )) is a rule in G . f ( k ) = sup { ∆( G ) : G ∈ G , ω ( G ) ≤ k } < R ( k − 1 , p − 1) ω ≥ k or α ≥ p G ∈ G with ω ( G ) ≤ k and ∆( G ) ≥ R ( k − 1 , p − 1) Vera Weil 9 / 11

Is the number of inclusionwise minimal rebels finite ? Let G be a hereditary graph class. For every p ∈ N such that K 1 , p is a rebel of G we find a monotone function f s.t. ∆( G ) ≤ f ( ω ( G )) is a rule in G . f ( k ) = sup { ∆( G ) : G ∈ G , ω ( G ) ≤ k } < R ( k − 1 , p − 1) ω ≥ k or α ≥ p G ∈ G with ω ( G ) ≤ k and ∆( G ) ≥ R ( k − 1 , p − 1) for some k ∈ N Vera Weil 9 / 11

Is the number of inclusionwise minimal rebels finite ? Let G be a hereditary graph class. For every p ∈ N such that K 1 , p is a rebel of G we find a monotone function f s.t. ∆( G ) ≤ f ( ω ( G )) is a rule in G . f ( k ) = sup { ∆( G ) : G ∈ G , ω ( G ) ≤ k } < R ( k − 1 , p − 1) ω ≥ k or α ≥ p G ∈ G with ω ( G ) ≤ k ∆ and ∆( G ) ≥ R ( k − 1 , p − 1) for some k ∈ N Vera Weil 9 / 11

Is the number of inclusionwise minimal rebels finite ? Let G be a hereditary graph class. For every p ∈ N such that K 1 , p is a rebel of G we find a monotone function f s.t. ∆( G ) ≤ f ( ω ( G )) is a rule in G . f ( k ) = sup { ∆( G ) : G ∈ G , ω ( G ) ≤ k } < R ( k − 1 , p − 1) ω ≥ k or α ≥ p G ∈ G with ω ( G ) ≤ k ω ≤ k − 1 ∆ and ∆( G ) ≥ R ( k − 1 , p − 1) for some k ∈ N Vera Weil 9 / 11