Maximum Matchings and Minimum Blocking Sets in 6 -Graphs Philipp - PowerPoint PPT Presentation

Known Results G ◦ ( P ) G △ ( P ) G ▽ ( P ) G � ( P ) G � ( P ) Half- Θ 6 -Graphs Θ 6 -Graphs Delaunay L ∞ -Delaunay t = 2.61 t = 2 t = 2 Spanning Ratio t 1.593 < t < 1.998

Known Results G ◦ ( P ) G △ ( P ) G ▽ ( P ) G � ( P ) G � ( P ) Half- Θ 6 -Graphs Θ 6 -Graphs Delaunay L ∞ -Delaunay t = 2.61 t = 2 t = 2 Spanning Ratio t 1.593 < t < 1.998 Max. Matching µ ( n )

Known Results G ◦ ( P ) G △ ( P ) G ▽ ( P ) G � ( P ) G � ( P ) Half- Θ 6 -Graphs Θ 6 -Graphs Delaunay L ∞ -Delaunay t = 2.61 t = 2 t = 2 Spanning Ratio t 1.593 < t < 1.998 µ ( n ) = ⌊ n Max. Matching µ ( n ) 2 ⌋

Known Results G ◦ ( P ) G △ ( P ) G ▽ ( P ) G � ( P ) G � ( P ) Half- Θ 6 -Graphs Θ 6 -Graphs Delaunay L ∞ -Delaunay t = 2.61 t = 2 t = 2 Spanning Ratio t 1.593 < t < 1.998 µ ( n ) = ⌊ n µ ( n ) = ⌊ n Max. Matching µ ( n ) 2 ⌋ 2 ⌋

Known Results G ◦ ( P ) G △ ( P ) G ▽ ( P ) G � ( P ) G � ( P ) Half- Θ 6 -Graphs Θ 6 -Graphs Delaunay L ∞ -Delaunay t = 2.61 t = 2 t = 2 Spanning Ratio t 1.593 < t < 1.998 µ ( n ) = ⌊ n µ ( n ) = ⌊ n Max. Matching µ ( n ) 2 ⌋ 2 ⌋ (even Hamil. path)

Known Results G ◦ ( P ) G △ ( P ) G ▽ ( P ) G � ( P ) G � ( P ) Half- Θ 6 -Graphs Θ 6 -Graphs Delaunay L ∞ -Delaunay t = 2.61 t = 2 t = 2 Spanning Ratio t 1.593 < t < 1.998 µ ( n ) = ⌈ n − 1 µ ( n ) = ⌊ n µ ( n ) = ⌊ n Max. Matching µ ( n ) 2 ⌋ 2 ⌋ 3 ⌉ (even Hamil. path)

Known Results G ◦ ( P ) G △ ( P ) G ▽ ( P ) G � ( P ) G � ( P ) Half- Θ 6 -Graphs Θ 6 -Graphs Delaunay L ∞ -Delaunay t = 2.61 t = 2 t = 2 Spanning Ratio t 1.593 < t < 1.998 µ ( n ) = ⌈ n − 1 µ ( n ) ≥ ⌈ n − 1 µ ( n ) = ⌊ n µ ( n ) = ⌊ n Max. Matching µ ( n ) 2 ⌋ 2 ⌋ 3 ⌉ 3 ⌉ (even Hamil. path)

Known Results G ◦ ( P ) G △ ( P ) G ▽ ( P ) G � ( P ) G � ( P ) Half- Θ 6 -Graphs Θ 6 -Graphs Delaunay L ∞ -Delaunay t = 2.61 t = 2 t = 2 Spanning Ratio t 1.593 < t < 1.998 µ ( n ) = ⌈ n − 1 µ ( n ) ≥ ⌈ n − 1 µ ( n ) = ⌊ n µ ( n ) = ⌊ n Max. Matching µ ( n ) 2 ⌋ 2 ⌋ 3 ⌉ 3 ⌉ (even µ ( n ) ≤ ⌊ n 2 ⌋ Hamil. path)

Known Results G ◦ ( P ) G △ ( P ) G ▽ ( P ) G � ( P ) G � ( P ) Half- Θ 6 -Graphs Θ 6 -Graphs Delaunay L ∞ -Delaunay t = 2.61 t = 2 t = 2 Spanning Ratio t 1.593 < t < 1.998 µ ( n ) = ⌈ n − 1 µ ( n ) ≥ ⌈ n − 1 µ ( n ) = ⌊ n µ ( n ) = ⌊ n Max. Matching µ ( n ) 2 ⌋ 2 ⌋ 3 ⌉ 3 ⌉ (even µ ( n ) ≤ ⌊ n 2 ⌋ Hamil. path)

The Tutte-Berge Formula G = ( V , E )

The Tutte-Berge Formula G = ( V , E ) S ⊆ V

The Tutte-Berge Formula G = ( V , E ) S ⊆ V

The Tutte-Berge Formula G = ( V , E ) S ⊆ V

The Tutte-Berge Formula G = ( V , E ) S ⊆ V comp ( G \ S )

The Tutte-Berge Formula G = ( V , E ) S ⊆ V comp ( G \ S )

The Tutte-Berge Formula G = ( V , E ) S ⊆ V comp ( G \ S ) odd ( G \ S )

The Tutte-Berge Formula G = ( V , E ) S ⊆ V comp ( G \ S ) odd ( G \ S ) [Tutte ’47] G has a (near-)perfect matching

The Tutte-Berge Formula G = ( V , E ) S ⊆ V comp ( G \ S ) odd ( G \ S ) [Tutte ’47] G has a (near-)perfect matching ⇔ odd ( S ) ≤ | S | for every S ⊆ V

The Tutte-Berge Formula G = ( V , E ) S ⊆ V comp ( G \ S ) odd ( G \ S ) [Tutte ’47] G has a (near-)perfect matching ⇔ odd ( S ) ≤ | S | for every S ⊆ V

The Tutte-Berge Formula G = ( V , E ) S ⊆ V comp ( G \ S ) odd ( G \ S ) [Tutte ’47] G has a (near-)perfect matching ⇔ odd ( S ) ≤ | S | for every S ⊆ V [Berge ’57]

The Tutte-Berge Formula G = ( V , E ) S ⊆ V comp ( G \ S ) odd ( G \ S ) [Tutte ’47] G has a (near-)perfect matching ⇔ odd ( S ) ≤ | S | for every S ⊆ V [Berge ’57] A maximum matching of G has size

The Tutte-Berge Formula G = ( V , E ) S ⊆ V comp ( G \ S ) odd ( G \ S ) [Tutte ’47] G has a (near-)perfect matching ⇔ odd ( S ) ≤ | S | for every S ⊆ V [Berge ’57] A maximum matching of G has size 1 2 ( | V | − max S ⊆ V ( odd ( G \ S ) − | S | ))

The Tutte-Berge Formula G = ( V , E ) S ⊆ V comp ( G \ S ) odd ( G \ S ) [Tutte ’47] G has a (near-)perfect matching ⇔ odd ( S ) ≤ | S | for every S ⊆ V [Berge ’57] A maximum matching of G has size 1 2 ( | V | − max S ⊆ V ( odd ( G \ S ) − | S | ))

The Tutte-Berge Formula G = ( V , E ) S ⊆ V comp ( G \ S ) odd ( G \ S ) [Tutte ’47] G has a (near-)perfect matching ⇔ odd ( S ) ≤ | S | for every S ⊆ V [Berge ’57] A maximum matching of G has size 1 2 ( | V | − max S ⊆ V ( odd ( G \ S ) − | S | ))

The Tutte-Berge Formula G = ( V , E ) S ⊆ V comp ( G \ S ) odd ( G \ S ) [Tutte ’47] G has a (near-)perfect matching ⇔ odd ( S ) ≤ | S | for every S ⊆ V [Berge ’57] A maximum matching of G has size 1 2 ( | V | − max S ⊆ V ( odd ( G \ S ) − | S | )) ⇒ there are max S ⊆ V ( odd ( G \ S ) − | S | ) unmatched vtcs

Upper Bound on Maximum Matching degree of a face

Upper Bound on Maximum Matching degree of a face d = 3

Upper Bound on Maximum Matching degree of a face d = 3

Upper Bound on Maximum Matching degree of a face d = 3

Upper Bound on Maximum Matching degree of a face d = 3 d = 6

Upper Bound on Maximum Matching degree of a face d = 3 d = 6

Upper Bound on Maximum Matching degree of a face d = 3 d = 6

Upper Bound on Maximum Matching degree of a face d = 3 d = 6 d = 11

Upper Bound on Maximum Matching ∑ d ≥ 3 ( d − 2 ) | F | △ deg d ≤ 2 n − 4 degree of a face d = 3 d = 6 d = 11

Upper Bound on Maximum Matching ∑ d ≥ 3 ( d − 2 ) | F | △ deg d ≤ 2 n − 4 G △ degree of a face d = 3 d = 6 d = 11

Upper Bound on Maximum Matching ∑ d ≥ 3 ( d − 2 ) | F | △ deg d ≤ 2 n − 4 G △ degree of a face d = 3 d = 6 d = 11

Upper Bound on Maximum Matching ∑ d ≥ 3 ( d − 2 ) | F | △ deg d ≤ 2 n − 4 G △ degree of a face d = 3 d = 6 d = 11

Upper Bound on Maximum Matching ∑ d ≥ 3 ( d − 2 ) | F | △ deg d ≤ 2 n − 4 G △ degree of a face d = 3 d = 6 d = 11

Upper Bound on Maximum Matching ∑ d ≥ 3 ( d − 2 ) | F | △ deg d ≤ 2 n − 4 G △ degree of a face d = 3 d = 6 d = 11

Upper Bound on Maximum Matching ∑ d ≥ 3 ( d − 2 ) | F | △ deg d ≤ 2 n − 4 G △ degree of a face A d = 3 d = 6 d = 11

Upper Bound on Maximum Matching ∑ d ≥ 3 ( d − 2 ) | F | △ deg d ≤ 2 n − 4 G △ degree of a face A d = 3 d = 6 d = 11

Upper Bound on Maximum Matching ∑ d ≥ 3 ( d − 2 ) | F | △ deg d ≤ 2 n − 4 G △ degree of a face A d = 3 d = 6 d = 11

Upper Bound on Maximum Matching ∑ d ≥ 3 ( d − 2 ) | F | △ deg d ≤ 2 n − 4 G △ degree of a face A G △ A d = 3 d = 6 d = 11