Tight Erds-Psa bounds for minors Jean-Florent Raymond (TU Berlin) - PowerPoint PPT Presentation

Tight Erdős-Pósa bounds for minors Jean-Florent Raymond (TU Berlin) Joint work with Wouter Cames van Batenburg, Tony Huynh, and Gwenaël Joret (Université Libre de Bruxelles).

Packing and covering in bipartite graphs Max. number of disjoint edges? pack K 2 3 Min. number of vertices to cover all edges? cover K 2 3 cover pack (Kőnig’s Theorem, 1931) Jean-Florent Raymond Tight Erdős-Pósa bounds for minors 2 / 18

Packing and covering in bipartite graphs Max. number of disjoint edges? pack K 2 3 Min. number of vertices to cover all edges? cover K 2 3 cover pack (Kőnig’s Theorem, 1931) Jean-Florent Raymond Tight Erdős-Pósa bounds for minors 2 / 18

Packing and covering in bipartite graphs Max. number of disjoint edges? pack K 2 3 Min. number of vertices to cover all edges? cover K 2 3 cover pack (Kőnig’s Theorem, 1931) Jean-Florent Raymond Tight Erdős-Pósa bounds for minors 2 / 18

Packing and covering in bipartite graphs Max. number of disjoint edges? pack K 2 3 Min. number of vertices to cover all edges? cover K 2 3 cover pack (Kőnig’s Theorem, 1931) Jean-Florent Raymond Tight Erdős-Pósa bounds for minors 2 / 18

Packing and covering in bipartite graphs Max. number of disjoint edges? Min. number of vertices to cover all edges? cover K 2 3 cover pack (Kőnig’s Theorem, 1931) Jean-Florent Raymond Tight Erdős-Pósa bounds for minors 2 / 18 pack K 2 = 3

Packing and covering in bipartite graphs Max. number of disjoint edges? Min. number of vertices to cover all edges? cover K 2 3 cover pack (Kőnig’s Theorem, 1931) Jean-Florent Raymond Tight Erdős-Pósa bounds for minors 2 / 18 pack K 2 = 3

Packing and covering in bipartite graphs Max. number of disjoint edges? Min. number of vertices to cover all edges? cover K 2 3 cover pack (Kőnig’s Theorem, 1931) Jean-Florent Raymond Tight Erdős-Pósa bounds for minors 2 / 18 pack K 2 = 3

Packing and covering in bipartite graphs Max. number of disjoint edges? Min. number of vertices to cover all edges? cover K 2 3 cover pack (Kőnig’s Theorem, 1931) Jean-Florent Raymond Tight Erdős-Pósa bounds for minors 2 / 18 pack K 2 = 3

Packing and covering in bipartite graphs Max. number of disjoint edges? Min. number of vertices to cover all edges? cover pack (Kőnig’s Theorem, 1931) Jean-Florent Raymond Tight Erdős-Pósa bounds for minors 2 / 18 pack K 2 = 3 cover K 2 = 3

Packing and covering in bipartite graphs Max. number of disjoint edges? Min. number of vertices to cover all edges? (Kőnig’s Theorem, 1931) Jean-Florent Raymond Tight Erdős-Pósa bounds for minors 2 / 18 pack K 2 = 3 cover K 2 = 3 cover = pack

Packing and covering cycles pack Tight Erdős-Pósa bounds for minors Jean-Florent Raymond (Erdős-Pósa Theorem, 1965) pack c pack cover 8 Max. number cover cycles to cover all cycles? Min. number of vertices 4 pack cycles of disjoint cycles? 3 / 18

Packing and covering cycles pack Tight Erdős-Pósa bounds for minors Jean-Florent Raymond (Erdős-Pósa Theorem, 1965) pack c pack cover 8 Max. number cover cycles to cover all cycles? Min. number of vertices 4 pack cycles of disjoint cycles? 3 / 18

Packing and covering cycles pack Tight Erdős-Pósa bounds for minors Jean-Florent Raymond (Erdős-Pósa Theorem, 1965) pack c pack cover 8 Max. number cover cycles to cover all cycles? Min. number of vertices 4 pack cycles of disjoint cycles? 3 / 18

Packing and covering cycles Max. number of disjoint cycles? Min. number of vertices to cover all cycles? cover cycles 8 pack cover c pack pack (Erdős-Pósa Theorem, 1965) Jean-Florent Raymond Tight Erdős-Pósa bounds for minors 3 / 18 pack cycles = 4

Packing and covering cycles Max. number of disjoint cycles? Min. number of vertices to cover all cycles? cover cycles 8 pack cover c pack pack (Erdős-Pósa Theorem, 1965) Jean-Florent Raymond Tight Erdős-Pósa bounds for minors 3 / 18 pack cycles = 4

Packing and covering cycles Max. number of disjoint cycles? Min. number of vertices to cover all cycles? cover cycles 8 pack cover c pack pack (Erdős-Pósa Theorem, 1965) Jean-Florent Raymond Tight Erdős-Pósa bounds for minors 3 / 18 pack cycles = 4

Packing and covering cycles Max. number of disjoint cycles? Min. number of vertices to cover all cycles? pack cover c pack pack (Erdős-Pósa Theorem, 1965) Jean-Florent Raymond Tight Erdős-Pósa bounds for minors 3 / 18 pack cycles = 4 cover cycles = 8

Packing and covering cycles Max. number of disjoint cycles? Min. number of vertices to cover all cycles? (Erdős-Pósa Theorem, 1965) Jean-Florent Raymond Tight Erdős-Pósa bounds for minors 3 / 18 pack cycles = 4 cover cycles = 8 pack ⩽ cover ⩽ c · pack log pack

The Erdős-Pósa Theorem Theorem (Erdős and Pósa, Can. J. Math. 1965) Every graph has one of the following: • k vertex-disjoint cycles; large packing vs. small cover Min-max theorem (like Kőnig’s and Menger’s theorems, etc.). Our goal: generalize from cycles to minor-models. Jean-Florent Raymond Tight Erdős-Pósa bounds for minors 4 / 18 • a feedback vertex set of size O ( k log k ) .

The Erdős-Pósa Theorem Theorem (Erdős and Pósa, Can. J. Math. 1965) Every graph has one of the following: • k vertex-disjoint cycles; large packing vs. small cover Min-max theorem (like Kőnig’s and Menger’s theorems, etc.). Our goal: generalize from cycles to minor-models. Jean-Florent Raymond Tight Erdős-Pósa bounds for minors 4 / 18 • a feedback vertex set of size O ( k log k ) .

The Erdős-Pósa Theorem Theorem (Erdős and Pósa, Can. J. Math. 1965) Every graph has one of the following: • k vertex-disjoint cycles; large packing vs. small cover Min-max theorem (like Kőnig’s and Menger’s theorems, etc.). Our goal: generalize from cycles to minor-models. Jean-Florent Raymond Tight Erdős-Pósa bounds for minors 4 / 18 • a feedback vertex set of size O ( k log k ) .

The Erdős-Pósa Theorem Theorem (Erdős and Pósa, Can. J. Math. 1965) Every graph has one of the following: • k vertex-disjoint cycles; large packing vs. small cover Min-max theorem (like Kőnig’s and Menger’s theorems, etc.). Our goal: generalize from cycles to minor-models. Jean-Florent Raymond Tight Erdős-Pósa bounds for minors 4 / 18 • a feedback vertex set of size O ( k log k ) .

Minor models Definition s.t. G H G has a H -model H is a minor of G Jean-Florent Raymond Tight Erdős-Pósa bounds for minors 5 / 18 An H -model in G is a set { S u } u ∈ V ( H ) of disjoint subsets of V ( G ) • the G [ S u ] ’s are connected; • edge uv in H ⇒ edge between S u and S v in G . ≽

Minor models Definition s.t. G H G has a H -model H is a minor of G Jean-Florent Raymond Tight Erdős-Pósa bounds for minors 5 / 18 An H -model in G is a set { S u } u ∈ V ( H ) of disjoint subsets of V ( G ) • the G [ S u ] ’s are connected; • edge uv in H ⇒ edge between S u and S v in G . ≽

Minor models Definition s.t. G H Jean-Florent Raymond Tight Erdős-Pósa bounds for minors 5 / 18 An H -model in G is a set { S u } u ∈ V ( H ) of disjoint subsets of V ( G ) • the G [ S u ] ’s are connected; • edge uv in H ⇒ edge between S u and S v in G . ≽ G has a H -model ⇐ ⇒ H is a minor of G

f is a gap of H . The Erdős-Pósa property of minor models Definition H has the Erdős-Pósa property if there is a function f s.t., for • G has k vertex-disjoint H -models; or Theorem (Robertson and Seymour, JCTB 1986) H has the Erdős-Pósa property iff H is planar. With which gap? Jean-Florent Raymond Tight Erdős-Pósa bounds for minors 6 / 18 every graph G and k ∈ N , • there is X ⊆ V ( G ) s.t. G − X is H -minor free and | X | ⩽ f ( k ) .

The Erdős-Pósa property of minor models Definition H has the Erdős-Pósa property if there is a function f s.t., for • G has k vertex-disjoint H -models; or Theorem (Robertson and Seymour, JCTB 1986) H has the Erdős-Pósa property iff H is planar. With which gap? Jean-Florent Raymond Tight Erdős-Pósa bounds for minors 6 / 18 every graph G and k ∈ N , • there is X ⊆ V ( G ) s.t. G − X is H -minor free and | X | ⩽ f ( k ) . f is a gap of H .

The Erdős-Pósa property of minor models Definition H has the Erdős-Pósa property if there is a function f s.t., for • G has k vertex-disjoint H -models; or Theorem (Robertson and Seymour, JCTB 1986) H has the Erdős-Pósa property iff H is planar. With which gap? Jean-Florent Raymond Tight Erdős-Pósa bounds for minors 6 / 18 every graph G and k ∈ N , • there is X ⊆ V ( G ) s.t. G − X is H -minor free and | X | ⩽ f ( k ) . f is a gap of H .

The Erdős-Pósa property of minor models Definition H has the Erdős-Pósa property if there is a function f s.t., for • G has k vertex-disjoint H -models; or Theorem (Robertson and Seymour, JCTB 1986) H has the Erdős-Pósa property iff H is planar. With which gap? Jean-Florent Raymond Tight Erdős-Pósa bounds for minors 6 / 18 every graph G and k ∈ N , • there is X ⊆ V ( G ) s.t. G − X is H -minor free and | X | ⩽ f ( k ) . f is a gap of H .