On (2 k , k )-connected graphs Zolt an Szigeti Combinatorial - PowerPoint PPT Presentation

On (2 k , k )-connected graphs Zolt´ an Szigeti Combinatorial Optimization Group Laboratoire G-SCOP INP Grenoble, France 11 septembre 2015 Joint work with : Olivier Durand de Gevigney Z. Szigeti (G-SCOP, Grenoble) On (2 k , k ) -connected graphs 11 septembre 2015 1 / 20

Outline Results on : Orientation Construction Splitting off Augmentation Concerning : Edge-connectivity (4 , 2)-connectivity (2 k , k )-connectivity Z. Szigeti (G-SCOP, Grenoble) On (2 k , k ) -connected graphs 11 septembre 2015 2 / 20

Orientation : arc-connectivity Definition 1 A digraph D is called k-arc-connected if ∀ ∅ � = X ⊂ V , | ρ D ( X ) | ≥ k . 2 A graph G is called k-edge-connected if ∀ ∅ � = X ⊂ V , d G ( X ) ≥ k . Z. Szigeti (G-SCOP, Grenoble) On (2 k , k ) -connected graphs 11 septembre 2015 3 / 20

Orientation : arc-connectivity Definition 1 A digraph D is called k-arc-connected if ∀ ∅ � = X ⊂ V , | ρ D ( X ) | ≥ k . 2 A graph G is called k-edge-connected if ∀ ∅ � = X ⊂ V , d G ( X ) ≥ k . Theorem (Nash-Williams) G has a k-arc-connected orientation if and only if G is 2 k-edge-connected. Z. Szigeti (G-SCOP, Grenoble) On (2 k , k ) -connected graphs 11 septembre 2015 3 / 20

Orientation : arc-connectivity Definition 1 A digraph D is called k-arc-connected if ∀ ∅ � = X ⊂ V , | ρ D ( X ) | ≥ k . 2 A graph G is called k-edge-connected if ∀ ∅ � = X ⊂ V , d G ( X ) ≥ k . Theorem (Nash-Williams) G has a k-arc-connected orientation if and only if G is 2 k-edge-connected. Necessity : k k X V − X � G Z. Szigeti (G-SCOP, Grenoble) On (2 k , k ) -connected graphs 11 septembre 2015 3 / 20

Orientation : arc-connectivity Definition 1 A digraph D is called k-arc-connected if ∀ ∅ � = X ⊂ V , | ρ D ( X ) | ≥ k . 2 A graph G is called k-edge-connected if ∀ ∅ � = X ⊂ V , d G ( X ) ≥ k . Theorem (Nash-Williams) G has a k-arc-connected orientation if and only if G is 2 k-edge-connected. Necessity : 2 k X V − X G Z. Szigeti (G-SCOP, Grenoble) On (2 k , k ) -connected graphs 11 septembre 2015 3 / 20

Orientation : k -vertex-connectivity Definition 1 A digraph D is called k-vertex-connected if | V | ≥ k + 1, ∀ X ⊂ V , | X | = k − 1 , D − X is 1-arc-connected. 2 A graph G is called k-vertex-connected if | V | ≥ k + 1, ∀ X ⊂ V , | X | = k − 1 , G − X is connected. Z. Szigeti (G-SCOP, Grenoble) On (2 k , k ) -connected graphs 11 septembre 2015 4 / 20

Orientation : k -vertex-connectivity Definition 1 A digraph D is called k-vertex-connected if | V | ≥ k + 1, ∀ X ⊂ V , | X | = k − 1 , D − X is 1-arc-connected. 2 A graph G is called k-vertex-connected if | V | ≥ k + 1, ∀ X ⊂ V , | X | = k − 1 , G − X is connected. Conjecture (Frank) G has a k-vertex-connected orientation if and only if | V | ≥ k + 1 and ∀ X ⊂ V , | X | < k, G − X is (2 k − 2 | X | ) -edge-connected. Z. Szigeti (G-SCOP, Grenoble) On (2 k , k ) -connected graphs 11 septembre 2015 4 / 20

Orientation : k -vertex-connectivity Definition 1 A digraph D is called k-vertex-connected if | V | ≥ k + 1, ∀ X ⊂ V , | X | = k − 1 , D − X is 1-arc-connected. 2 A graph G is called k-vertex-connected if | V | ≥ k + 1, ∀ X ⊂ V , | X | = k − 1 , G − X is connected. Conjecture (Frank) G has a k-vertex-connected orientation if and only if | V | ≥ k + 1 and ∀ X ⊂ V , | X | < k, G − X is (2 k − 2 | X | ) -edge-connected. Theorem (Durand de Gevigney) ( k ≥ 3) 1 This conjecture is false. Z. Szigeti (G-SCOP, Grenoble) On (2 k , k ) -connected graphs 11 septembre 2015 4 / 20

Orientation : k -vertex-connectivity Definition 1 A digraph D is called k-vertex-connected if | V | ≥ k + 1, ∀ X ⊂ V , | X | = k − 1 , D − X is 1-arc-connected. 2 A graph G is called k-vertex-connected if | V | ≥ k + 1, ∀ X ⊂ V , | X | = k − 1 , G − X is connected. Conjecture (Frank) G has a k-vertex-connected orientation if and only if | V | ≥ k + 1 and ∀ X ⊂ V , | X | < k, G − X is (2 k − 2 | X | ) -edge-connected. Theorem (Durand de Gevigney) ( k ≥ 3) 1 This conjecture is false. 2 Deciding whether G has a k-vertex-connected orientation is NP-complete. Z. Szigeti (G-SCOP, Grenoble) On (2 k , k ) -connected graphs 11 septembre 2015 4 / 20

Counter-example for k = 3 Example of Durand de Gevigney Z. Szigeti (G-SCOP, Grenoble) On (2 k , k ) -connected graphs 11 septembre 2015 5 / 20

Orientation : 2-vertex-connectivity Remark (Necessary condition) Example If � G is 2-vertex-connected, then Z. Szigeti (G-SCOP, Grenoble) On (2 k , k ) -connected graphs 11 septembre 2015 6 / 20

Orientation : 2-vertex-connectivity Remark (Necessary condition) Example If � G is 2-vertex-connected, then | V | ≥ 3, Z. Szigeti (G-SCOP, Grenoble) On (2 k , k ) -connected graphs 11 septembre 2015 6 / 20

Orientation : 2-vertex-connectivity Remark (Necessary condition) Example If � G is 2-vertex-connected, then | V | ≥ 3, 1 G is 4-edge-connected and, Z. Szigeti (G-SCOP, Grenoble) On (2 k , k ) -connected graphs 11 septembre 2015 6 / 20

Orientation : 2-vertex-connectivity Remark (Necessary condition) Example If � G is 2-vertex-connected, then | V | ≥ 3, 1 G is 4-edge-connected and, 2 for all v ∈ V , G − v is 2-edge-connected. Z. Szigeti (G-SCOP, Grenoble) On (2 k , k ) -connected graphs 11 septembre 2015 6 / 20

Orientation : 2-vertex-connectivity Definition Example A graph G is called (4 , 2) -connected if | V | ≥ 3, 1 G is 4-edge-connected and, 2 for all v ∈ V , G − v is 2-edge-connected. Z. Szigeti (G-SCOP, Grenoble) On (2 k , k ) -connected graphs 11 septembre 2015 6 / 20

Orientation : 2-vertex-connectivity Definition Example A graph G is called (4 , 2) -connected if | V | ≥ 3, 1 G is 4-edge-connected and, 2 for all v ∈ V , G − v is 2-edge-connected. Theorem (Sufficent condition) A graph G has a 2 -vertex-connected orientation 1 if G is (4 , 2) -connected and Eulerian (Berg, Jord´ an). Z. Szigeti (G-SCOP, Grenoble) On (2 k , k ) -connected graphs 11 septembre 2015 6 / 20

Orientation : 2-vertex-connectivity Definition Example A graph G is called (4 , 2) -connected if | V | ≥ 3, 1 G is 4-edge-connected and, 2 for all v ∈ V , G − v is 2-edge-connected. Theorem (Sufficent condition) A graph G has a 2 -vertex-connected orientation 1 if G is (4 , 2) -connected and Eulerian (Berg, Jord´ an). 2 if G is 18 -vertex-connected (Jord´ an). Z. Szigeti (G-SCOP, Grenoble) On (2 k , k ) -connected graphs 11 septembre 2015 6 / 20

Orientation : 2-vertex-connectivity Definition Example A graph G is called (4 , 2) -connected if | V | ≥ 3, 1 G is 4-edge-connected and, 2 for all v ∈ V , G − v is 2-edge-connected. Theorem (Sufficent condition) A graph G has a 2 -vertex-connected orientation 1 if G is (4 , 2) -connected and Eulerian (Berg, Jord´ an). 2 if G is 18 -vertex-connected (Jord´ an). 3 if G is 14 -vertex-connected (Cheriyan, Durand de Gevigney, Szigeti). Z. Szigeti (G-SCOP, Grenoble) On (2 k , k ) -connected graphs 11 septembre 2015 6 / 20

Orientation : 2-vertex-connectivity Definition Example A graph G is called (4 , 2) -connected if | V | ≥ 3, 1 G is 4-edge-connected and, 2 for all v ∈ V , G − v is 2-edge-connected. Theorem (Sufficent condition) A graph G has a 2 -vertex-connected orientation 1 if G is (4 , 2) -connected and Eulerian (Berg, Jord´ an). 2 if G is 18 -vertex-connected (Jord´ an). 3 if G is 14 -vertex-connected (Cheriyan, Durand de Gevigney, Szigeti). Theorem (Thomassen) G has a 2 -vertex-connected orientation if and only if G is (4 , 2) -connected. Z. Szigeti (G-SCOP, Grenoble) On (2 k , k ) -connected graphs 11 septembre 2015 6 / 20

Construction : edge-connectivity Theorem (Lov´ asz) Example A graph is 2 k-edge-connected if and only if it can be obtained from K 2 k by a sequence of 2 the following two operations : (a) adding a new edge, (b) pinching k edges. Z. Szigeti (G-SCOP, Grenoble) On (2 k , k ) -connected graphs 11 septembre 2015 7 / 20

Construction : edge-connectivity Theorem (Lov´ asz) Example A graph is 2 k-edge-connected if and only if it can be obtained from K 2 k by a sequence of 2 the following two operations : (a) adding a new edge, (b) pinching k edges. Z. Szigeti (G-SCOP, Grenoble) On (2 k , k ) -connected graphs 11 septembre 2015 7 / 20

Construction : edge-connectivity Theorem (Lov´ asz) Example A graph is 2 k-edge-connected if and only if it can be obtained from K 2 k by a sequence of 2 the following two operations : (a) adding a new edge, (b) pinching k edges. Z. Szigeti (G-SCOP, Grenoble) On (2 k , k ) -connected graphs 11 septembre 2015 7 / 20

Construction : edge-connectivity Theorem (Lov´ asz) Example A graph is 2 k-edge-connected if and only if it can be obtained from K 2 k by a sequence of 2 the following two operations : (a) adding a new edge, (b) pinching k edges. Z. Szigeti (G-SCOP, Grenoble) On (2 k , k ) -connected graphs 11 septembre 2015 7 / 20

Construction : edge-connectivity Theorem (Lov´ asz) Example A graph is 2 k-edge-connected if and only if it can be obtained from K 2 k by a sequence of 2 the following two operations : (a) adding a new edge, (b) pinching k edges. Z. Szigeti (G-SCOP, Grenoble) On (2 k , k ) -connected graphs 11 septembre 2015 7 / 20