Asymptotic growth of minor-closed classes of graphs Olivier - PowerPoint PPT Presentation

Asymptotic growth of minor-closed classes of graphs Olivier Bernardi - Centre de Recerca Matemàtica Joint work with Marc Noy and Dominic Welsh Humboldt-Universität zu Berlin, June 2007 Berlin, June 2007 Olivier Bernardi – p.1/22

Growth of minor-closed classes Graph minors. Rough classification (Polynomial, Exponential, Factorial) and refinements. Growth constants. Berlin, June 2007 Olivier Bernardi – p.2/22

Graph minors Berlin, June 2007 Olivier Bernardi – p.3/22

Deletion and contraction Deletion Contraction Berlin, June 2007 Olivier Bernardi – p.4/22

Minors A graph H is minor of a graph G if H can be obtained from G by a sequence of deletions and contractions. Example: H ≺ G Berlin, June 2007 ▽ Olivier Bernardi – p.5/22

Minors A graph H is minor of a graph G if H can be obtained from G by a sequence of deletions and contractions. Example: H ≺ G G Berlin, June 2007 ▽ Olivier Bernardi – p.5/22

Minors A graph H is minor of a graph G if H can be obtained from G by a sequence of deletions and contractions. Example: H ≺ G H G Berlin, June 2007 Olivier Bernardi – p.5/22

Minor-closed classes A class of graphs is a set of labeled graphs closed under isomorphism. Berlin, June 2007 ▽ Olivier Bernardi – p.6/22

Minor-closed classes A class of graphs is a set of labeled graphs closed under isomorphism. A class of graphs is closed under minors if it is closed under deletions and contractions. Example: Forests: Planar graphs: Berlin, June 2007 ▽ Olivier Bernardi – p.6/22

Minor-closed classes A class of graphs is a set of labeled graphs closed under isomorphism. A class of graphs is closed under minors if it is closed under deletions and contractions. A graph G is a minimal excluded minor for a class G if G is not in G but any proper minor of G is in G . Example: Forests: Planar graphs: Berlin, June 2007 ▽ Olivier Bernardi – p.6/22

Minor-closed classes A class of graphs is a set of labeled graphs closed under isomorphism. A class of graphs is closed under minors if it is closed under deletions and contractions. A graph G is a minimal excluded minor for a class G if G is not in G but any proper minor of G is in G . Minor theorem [Robertson and Seymour]: For any minor-closed class, the number of minimal excluded minors is finite. Berlin, June 2007 Olivier Bernardi – p.6/22

Growth of minor-closed classes: a classification Berlin, June 2007 Olivier Bernardi – p.7/22

Upper-bound g n : number of graphs with n vertices in G . Berlin, June 2007 ▽ Olivier Bernardi – p.8/22

Upper-bound g n : number of graphs with n vertices in G . Theorem [Norine, Seymour, Thomas, Wallan]: For any proper minor-closed class G , there exists c such that g n ≤ c n n ! . Berlin, June 2007 Olivier Bernardi – p.8/22

Classification Theorem [Bernardi, Noy, Welsh]: The growth of a proper minor-closed class G , is either n ! ≤ g n ≤ c n n ! factorial for some c > 1 , n !! ≤ g n < ǫ n n ! pseudo-factorial for all ǫ > 0 , 2 n ≤ g n ≤ c n exponential for some c > 2 , n ( n − 1) ≤ g n ≤ n c polynomial for some c ≥ 2 , 2 or constant g n = 0 or 1 . Berlin, June 2007 Olivier Bernardi – p.9/22

Classification : proof The growth of a proper minor-closed class G , is either n ! ≤ g n ≤ c n n ! for some c > 1 , factorial Path: n !! ≤ g n < ǫ n n ! for all ǫ > 0 , pseudo-factorial matching: 2 n ≤ g n ≤ c n exponential for some c > 2 , star: ≤ g n ≤ n c for some c ≥ 2 , n ( n − 1) polynomial 2 edge: g n = 0 or 1 . or constant Berlin, June 2007 Olivier Bernardi – p.10/22

Proof : path obstruction Prop: If G contains all paths, g n ≥ n ! , then otherwise g n < ǫ n n ! for all ǫ > 0 . Berlin, June 2007 ▽ Olivier Bernardi – p.11/22

Proof : path obstruction Lemma: The class G = Ex ( P k ) satisfies g n < ǫ n n ! for all ǫ > 0 . Proof: • The class ¯ g n < ǫ n n ! for all ǫ > 0 . G = Ex ( P k , ∆) satisfies ¯ < k G ( z ) ≤ F ( z ) = e ze ze...zez ¯ Berlin, June 2007 ▽ Olivier Bernardi – p.11/22

Proof : path obstruction Lemma: The class G = Ex ( P k ) satisfies g n < ǫ n n ! for all ǫ > 0 . Proof: • The class ¯ g n < ǫ n n ! for all ǫ > 0 . G = Ex ( P k , ∆) satisfies ¯ < k • The class G = Ex ( P k ) satisfies g n < 2 kn ¯ g n . < k Berlin, June 2007 Olivier Bernardi – p.11/22

Proof : matching obstruction Prop: If G contains all matchings, then g n ≥ ( n − 1)!! = ( n − 1)( n − 3) · · · , otherwise g n < c n for some c > 0 . Berlin, June 2007 ▽ Olivier Bernardi – p.12/22

Proof : matching obstruction Prop: If G contains all matchings, then g n ≥ ( n − 1)!! = ( n − 1)( n − 3) · · · , otherwise g n < c n for some c > 0 . Lemma: The class G = Ex ( M k ) satisfies g n < P ( n ) · 2 2 kn . . . . Berlin, June 2007 Olivier Bernardi – p.12/22

Proof : star obstruction Prop: If G contains all stars, g n ≥ 2 n , then otherwise g n < n c for some c > 0 . Berlin, June 2007 ▽ Olivier Bernardi – p.13/22

Proof : star obstruction Prop: If G contains all stars, g n ≥ 2 n , then otherwise g n < n c for some c > 0 . Lemma: The class G = Ex ( M k , S k ) satisfies g n < P ( n ) . . . . Berlin, June 2007 Olivier Bernardi – p.13/22

Proof : edge obstruction Prop: If G contains the graphs with 1 edge, g n ≥ n ( n − 1) then , 2 otherwise g n = 0 or 1 for large n . Berlin, June 2007 Olivier Bernardi – p.14/22

Refined classification Theorem: The growth of a minor-closed class G , is either n ! ≤ g n ≤ c n n ! for some c > 1 , factorial pseudo-factorial n !! ≤ g n < ǫ n n ! for all ǫ > 0 , 2 n ≤ g n ≤ c n exponential for some c > 2 , n ( n − 1) ≤ g n ≤ n c polynomial for some c ≥ 2 , 2 or constant g n = 0 or 1 . Berlin, June 2007 ▽ Olivier Bernardi – p.15/22

Refined classification Theorem: The growth of a minor-closed class G , is either n ! ≤ g n ≤ c n n ! for some c > 1 , factorial pseudo-factorial n !! ≤ g n < ǫ n n ! for all ǫ > 0 , 2 n ≤ g n ≤ c n exponential for some c > 2 , n ( n − 1) ≤ g n ≤ n c polynomial for some c ≥ 2 , 2 or constant g n = 0 or 1 . Theorem : If G has polynomial growth, then there is a polynomial P such that g n = P ( n ) for large n . Berlin, June 2007 ▽ Olivier Bernardi – p.15/22

Refined classification Theorem: The growth of a minor-closed class G , is either n ! ≤ g n ≤ c n n ! for some c > 1 , factorial pseudo-factorial n !! ≤ g n < ǫ n n ! for all ǫ > 0 , 2 n ≤ g n ≤ c n exponential for some c > 2 , n ( n − 1) ≤ g n ≤ n c polynomial for some c ≥ 2 , 2 or constant g n = 0 or 1 . Theorem : If G has exponential growth, then there is an integer k ≥ 2 such that g n ≍ poly k n . Berlin, June 2007 ▽ Olivier Bernardi – p.15/22

Refined classification Theorem: The growth of a minor-closed class G , is either n ! ≤ g n ≤ c n n ! for some c > 1 , factorial n !! ≤ g n < ǫ n n ! pseudo-factorial for all ǫ > 0 , 2 n ≤ g n ≤ c n exponential for some c > 2 , n ( n − 1) ≤ g n ≤ n c polynomial for some c ≥ 2 , 2 or constant g n = 0 or 1 . Theorem : If G has pseudo-factorial growth, then either n (1 − δ ) n ≤ g n < ǫ n n n for all δ, ǫ > 0 , ( k − 1) n . or there is an integer k such that g n ≍ exp n k Berlin, June 2007 Olivier Bernardi – p.15/22

Growth constants Berlin, June 2007 Olivier Bernardi – p.16/22

Growth constants � g n � 1 /n The growth constant of a class G is γ ( G ) = lim sup . n ! We are interested in the set Γ = { γ ( G ) / G minor-closed } . Berlin, June 2007 ▽ Olivier Bernardi – p.17/22

Growth constants � g n � 1 /n The growth constant of a class G is γ ( G ) = lim sup . n ! Example: • G = Ex ( P k ) : growth constant 0 . • Path forests: growth constant 1 . • Caterpillar forests: growth constant ξ ≈ 1 . 76 , root of e 1 /x = x . • Forests: growth constant e ≈ 2 . 71 . Berlin, June 2007 Olivier Bernardi – p.17/22

Class Growth Reference Ex ( P k ) 0 Path forests 1 Standard ξ ≈ 1 . 76 Caterpillars Forests = Ex ( K 3 ) e ≈ 2 . 71 Standard Ex ( C 4 ) 3 . 63 Ex ( K 4 − e ) 4 . 18 Ex ( C 5 ) 4 . 60 Outer-planar = Ex ( K 4 , K 2 , 3 ) 7 . 320 [Bodirsky et al. ] Ex ( K 2 , 3 ) 7 . 327 [Bodirsky et al. ] Series parallel = Ex ( K 4 ) 9 . 07 [Bodirsky et al. ] Ex ( W 4 ) 11 . 54 [Gimenez et al. ] Ex ( K 5 − e ) 12 . 96 [Gimenez et al. ] Ex ( K 2 × K 3 ) 14 . 13 [Gimenez et al. ] Planar 27 . 226 [Gimenez & Noy] Embed. in fixed surface 27 . 226 [McDiarmid] Ex ( K 3 , 3 ) 27 . 229 [Gerke et al. ] Berlin, June 2007 Olivier Bernardi – p.18/22

Apex construction (McDiarmid) ξ e 0 1 Berlin, June 2007 ▽ Olivier Bernardi – p.19/22

Apex construction (McDiarmid) ξ e 2 ξ 2 e 0 1 2 4 Proposition: If γ is a growth constant, then 2 γ also. Proof: γ ( G A ) = 2 · γ ( G ) . Berlin, June 2007 Olivier Bernardi – p.19/22

Gaps ξ e 2 ξ 2 e 0 1 2 4 Berlin, June 2007 ▽ Olivier Bernardi – p.20/22

Gaps ξ e 2 ξ 2 e 0 1 2 4 Proposition: There is no growth constant between 1 and ξ . Proof: Obstructions : Berlin, June 2007 Olivier Bernardi – p.20/22