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

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

Mar 20, 2024 346 likes •2.11k views

Maximum Matchings and Minimum Blocking Sets in 6 -Graphs Philipp Kindermann Universit at W urzburg Therese Biedl Ahmad Biniaz Veronika Irvine Kshitij Jain Anna Lubiw Proximity Graphs P : set of n points in the plane Proximity

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SLIDE 1

Maximum Matchings and Minimum Blocking Sets in Θ6-Graphs

Philipp Kindermann Universit¨ at W¨ urzburg Therese Biedl Ahmad Biniaz Veronika Irvine Kshitij Jain Anna Lubiw

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SLIDE 2

Proximity Graphs

P: set of n points in the plane

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SLIDE 3

Proximity Graphs

P: set of n points in the plane

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SLIDE 4

Proximity Graphs

P: set of n points in the plane

S: set of geom. objects in the plane

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SLIDE 5

Proximity Graphs

P: set of n points in the plane

S: set of geom. objects in the plane

GS(P)=(P, E):

(p, q) ∈ E ⇔ ∃S ∈ S : S ∩ P = {p, q}

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SLIDE 6

Proximity Graphs

P: set of n points in the plane

S: set of geom. objects in the plane

GS(P)=(P, E):

(p, q) ∈ E ⇔ ∃S ∈ S : S ∩ P = {p, q}

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SLIDE 7

Proximity Graphs

P: set of n points in the plane

S: set of geom. objects in the plane

GS(P)=(P, E):

(p, q) ∈ E ⇔ ∃S ∈ S : S ∩ P = {p, q}

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SLIDE 8

Proximity Graphs

P: set of n points in the plane

S: set of geom. objects in the plane

GS(P)=(P, E):

(p, q) ∈ E ⇔ ∃S ∈ S : S ∩ P = {p, q}

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SLIDE 9

Proximity Graphs

P: set of n points in the plane

S: set of geom. objects in the plane

GS(P)=(P, E):

(p, q) ∈ E ⇔ ∃S ∈ S : S ∩ P = {p, q}

G◦(P)

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SLIDE 10

Proximity Graphs

P: set of n points in the plane

S: set of geom. objects in the plane

GS(P)=(P, E):

(p, q) ∈ E ⇔ ∃S ∈ S : S ∩ P = {p, q}

G◦(P)

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SLIDE 11

Proximity Graphs

P: set of n points in the plane

S: set of geom. objects in the plane

GS(P)=(P, E):

(p, q) ∈ E ⇔ ∃S ∈ S : S ∩ P = {p, q}

G◦(P)

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SLIDE 12

Proximity Graphs

P: set of n points in the plane

S: set of geom. objects in the plane

GS(P)=(P, E):

(p, q) ∈ E ⇔ ∃S ∈ S : S ∩ P = {p, q}

G◦(P)

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SLIDE 13

Proximity Graphs

P: set of n points in the plane

S: set of geom. objects in the plane

GS(P)=(P, E):

(p, q) ∈ E ⇔ ∃S ∈ S : S ∩ P = {p, q}

G◦(P)

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SLIDE 14

Proximity Graphs

P: set of n points in the plane

S: set of geom. objects in the plane

GS(P)=(P, E):

(p, q) ∈ E ⇔ ∃S ∈ S : S ∩ P = {p, q}

G◦(P)

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SLIDE 15

Proximity Graphs

P: set of n points in the plane

S: set of geom. objects in the plane

GS(P)=(P, E):

(p, q) ∈ E ⇔ ∃S ∈ S : S ∩ P = {p, q}

G◦(P)

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SLIDE 16

Proximity Graphs

P: set of n points in the plane

S: set of geom. objects in the plane

GS(P)=(P, E):

(p, q) ∈ E ⇔ ∃S ∈ S : S ∩ P = {p, q}

G◦(P)

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SLIDE 17

Proximity Graphs

P: set of n points in the plane

S: set of geom. objects in the plane

GS(P)=(P, E):

(p, q) ∈ E ⇔ ∃S ∈ S : S ∩ P = {p, q}

G◦(P)

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SLIDE 18

Proximity Graphs

P: set of n points in the plane

S: set of geom. objects in the plane

GS(P)=(P, E):

(p, q) ∈ E ⇔ ∃S ∈ S : S ∩ P = {p, q}

G◦(P)

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SLIDE 19

Proximity Graphs

P: set of n points in the plane

S: set of geom. objects in the plane

GS(P)=(P, E):

(p, q) ∈ E ⇔ ∃S ∈ S : S ∩ P = {p, q}

G◦(P) Delaunay

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SLIDE 20

Proximity Graphs

P: set of n points in the plane

S: set of geom. objects in the plane

GS(P)=(P, E):

(p, q) ∈ E ⇔ ∃S ∈ S : S ∩ P = {p, q}

G◦(P) Delaunay G(P)

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SLIDE 21

Proximity Graphs

P: set of n points in the plane

S: set of geom. objects in the plane

GS(P)=(P, E):

(p, q) ∈ E ⇔ ∃S ∈ S : S ∩ P = {p, q}

G◦(P) Delaunay G(P)

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SLIDE 22

Proximity Graphs

P: set of n points in the plane

S: set of geom. objects in the plane

GS(P)=(P, E):

(p, q) ∈ E ⇔ ∃S ∈ S : S ∩ P = {p, q}

G◦(P) Delaunay G(P)

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SLIDE 23

Proximity Graphs

P: set of n points in the plane

S: set of geom. objects in the plane

GS(P)=(P, E):

(p, q) ∈ E ⇔ ∃S ∈ S : S ∩ P = {p, q}

G◦(P) Delaunay G(P)

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SLIDE 24

Proximity Graphs

P: set of n points in the plane

S: set of geom. objects in the plane

GS(P)=(P, E):

(p, q) ∈ E ⇔ ∃S ∈ S : S ∩ P = {p, q}

G◦(P) Delaunay G(P)

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SLIDE 25

Proximity Graphs

P: set of n points in the plane

S: set of geom. objects in the plane

GS(P)=(P, E):

(p, q) ∈ E ⇔ ∃S ∈ S : S ∩ P = {p, q}

G◦(P) Delaunay G(P)

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SLIDE 26

Proximity Graphs

P: set of n points in the plane

S: set of geom. objects in the plane

GS(P)=(P, E):

(p, q) ∈ E ⇔ ∃S ∈ S : S ∩ P = {p, q}

G◦(P) Delaunay G(P)

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SLIDE 27

Proximity Graphs

P: set of n points in the plane

S: set of geom. objects in the plane

GS(P)=(P, E):

(p, q) ∈ E ⇔ ∃S ∈ S : S ∩ P = {p, q}

G◦(P) Delaunay G(P) L∞-Delaunay

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SLIDE 28

Proximity Graphs

P: set of n points in the plane

S: set of geom. objects in the plane

GS(P)=(P, E):

(p, q) ∈ E ⇔ ∃S ∈ S : S ∩ P = {p, q}

G◦(P) Delaunay G(P) L∞-Delaunay G△(P)

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SLIDE 29

Proximity Graphs

P: set of n points in the plane

S: set of geom. objects in the plane

GS(P)=(P, E):

(p, q) ∈ E ⇔ ∃S ∈ S : S ∩ P = {p, q}

G◦(P) Delaunay G(P) L∞-Delaunay G△(P)

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SLIDE 30

Proximity Graphs

P: set of n points in the plane

S: set of geom. objects in the plane

GS(P)=(P, E):

(p, q) ∈ E ⇔ ∃S ∈ S : S ∩ P = {p, q}

G◦(P) Delaunay G(P) L∞-Delaunay G△(P)

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SLIDE 31

Proximity Graphs

P: set of n points in the plane

S: set of geom. objects in the plane

GS(P)=(P, E):

(p, q) ∈ E ⇔ ∃S ∈ S : S ∩ P = {p, q}

G◦(P) Delaunay G(P) L∞-Delaunay G△(P)

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SLIDE 32

Proximity Graphs

P: set of n points in the plane

S: set of geom. objects in the plane

GS(P)=(P, E):

(p, q) ∈ E ⇔ ∃S ∈ S : S ∩ P = {p, q}

G◦(P) Delaunay G(P) L∞-Delaunay G△(P)

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SLIDE 33

Proximity Graphs

P: set of n points in the plane

S: set of geom. objects in the plane

GS(P)=(P, E):

(p, q) ∈ E ⇔ ∃S ∈ S : S ∩ P = {p, q}

G◦(P) Delaunay G(P) L∞-Delaunay G△(P) G▽(P)

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SLIDE 34

Proximity Graphs

P: set of n points in the plane

S: set of geom. objects in the plane

GS(P)=(P, E):

(p, q) ∈ E ⇔ ∃S ∈ S : S ∩ P = {p, q}

G◦(P) Delaunay G(P) L∞-Delaunay G△(P) G▽(P)

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SLIDE 35

Proximity Graphs

P: set of n points in the plane

S: set of geom. objects in the plane

GS(P)=(P, E):

(p, q) ∈ E ⇔ ∃S ∈ S : S ∩ P = {p, q}

G◦(P) Delaunay G(P) L∞-Delaunay G△(P) G▽(P)

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SLIDE 36

Proximity Graphs

P: set of n points in the plane

S: set of geom. objects in the plane

GS(P)=(P, E):

(p, q) ∈ E ⇔ ∃S ∈ S : S ∩ P = {p, q}

G◦(P) Delaunay G(P) L∞-Delaunay G△(P) G▽(P)

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SLIDE 37

Proximity Graphs

P: set of n points in the plane

S: set of geom. objects in the plane

GS(P)=(P, E):

(p, q) ∈ E ⇔ ∃S ∈ S : S ∩ P = {p, q}

G◦(P) Delaunay G(P) L∞-Delaunay G△(P) G▽(P)

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SLIDE 38

Proximity Graphs

P: set of n points in the plane

S: set of geom. objects in the plane

GS(P)=(P, E):

(p, q) ∈ E ⇔ ∃S ∈ S : S ∩ P = {p, q}

G◦(P) Delaunay G(P) L∞-Delaunay G△(P) G▽(P)

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SLIDE 39

Proximity Graphs

P: set of n points in the plane

S: set of geom. objects in the plane

GS(P)=(P, E):

(p, q) ∈ E ⇔ ∃S ∈ S : S ∩ P = {p, q}

G◦(P) Delaunay G(P) L∞-Delaunay G△(P) G▽(P) Half-Θ6-Graphs

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SLIDE 40

Proximity Graphs

P: set of n points in the plane

S: set of geom. objects in the plane

GS(P)=(P, E):

(p, q) ∈ E ⇔ ∃S ∈ S : S ∩ P = {p, q}

G◦(P) Delaunay G(P) L∞-Delaunay G△(P) G▽(P) Half-Θ6-Graphs G(P) Θ6-Graph

∪

=

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SLIDE 41
  • Alt. Definition of Θ6-Graphs
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SLIDE 42
  • Alt. Definition of Θ6-Graphs

G△(P)

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SLIDE 43
  • Alt. Definition of Θ6-Graphs

G△(P)

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SLIDE 44
  • Alt. Definition of Θ6-Graphs

G△(P)

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SLIDE 45
  • Alt. Definition of Θ6-Graphs

G△(P)

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SLIDE 46
  • Alt. Definition of Θ6-Graphs

G△(P) G▽(P)

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SLIDE 47
  • Alt. Definition of Θ6-Graphs

G△(P) G▽(P)

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SLIDE 48
  • Alt. Definition of Θ6-Graphs

G△(P) G▽(P)

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SLIDE 49
  • Alt. Definition of Θ6-Graphs

G△(P) G▽(P)

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SLIDE 50
  • Alt. Definition of Θ6-Graphs

G△(P) G▽(P)

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SLIDE 51
  • Alt. Definition of Θ6-Graphs

G△(P) G▽(P)

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SLIDE 52
  • Alt. Definition of Θ6-Graphs

G△(P) G▽(P) G(P)

∪

=

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SLIDE 53

Known Results

G◦(P) Delaunay G(P) L∞-Delaunay G△(P) G▽(P) Half-Θ6-Graphs G(P) Θ6-Graphs

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SLIDE 54

Known Results

G◦(P) Delaunay G(P) L∞-Delaunay G△(P) G▽(P) Half-Θ6-Graphs G(P) Θ6-Graphs

Spanning Ratio t

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SLIDE 55

Known Results

G◦(P) Delaunay G(P) L∞-Delaunay G△(P) G▽(P) Half-Θ6-Graphs G(P) Θ6-Graphs

Spanning Ratio t 1.593<t<1.998

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SLIDE 56

Known Results

G◦(P) Delaunay G(P) L∞-Delaunay G△(P) G▽(P) Half-Θ6-Graphs G(P) Θ6-Graphs

Spanning Ratio t 1.593<t<1.998 t = 2.61

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SLIDE 57

Known Results

G◦(P) Delaunay G(P) L∞-Delaunay G△(P) G▽(P) Half-Θ6-Graphs G(P) Θ6-Graphs

Spanning Ratio t 1.593<t<1.998 t = 2 t = 2.61

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SLIDE 58

Known Results

G◦(P) Delaunay G(P) L∞-Delaunay G△(P) G▽(P) Half-Θ6-Graphs G(P) Θ6-Graphs

Spanning Ratio t t = 2 1.593<t<1.998 t = 2 t = 2.61

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SLIDE 59

Known Results

G◦(P) Delaunay G(P) L∞-Delaunay G△(P) G▽(P) Half-Θ6-Graphs G(P) Θ6-Graphs

  • Max. Matching µ(n)

Spanning Ratio t t = 2 1.593<t<1.998 t = 2 t = 2.61

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SLIDE 60

Known Results

G◦(P) Delaunay G(P) L∞-Delaunay G△(P) G▽(P) Half-Θ6-Graphs G(P) Θ6-Graphs

  • Max. Matching µ(n)

Spanning Ratio t t = 2 1.593<t<1.998 t = 2 t = 2.61 µ(n) = ⌊n

2⌋

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SLIDE 61

Known Results

G◦(P) Delaunay G(P) L∞-Delaunay G△(P) G▽(P) Half-Θ6-Graphs G(P) Θ6-Graphs

  • Max. Matching µ(n)

Spanning Ratio t t = 2 1.593<t<1.998 t = 2 t = 2.61 µ(n) = ⌊n

2⌋

µ(n) = ⌊n

2⌋

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SLIDE 62

Known Results

G◦(P) Delaunay G(P) L∞-Delaunay G△(P) G▽(P) Half-Θ6-Graphs G(P) Θ6-Graphs

  • Max. Matching µ(n)

Spanning Ratio t t = 2 1.593<t<1.998 t = 2 t = 2.61 µ(n) = ⌊n

2⌋

µ(n) = ⌊n

2⌋

(even

  • Hamil. path)
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SLIDE 63

Known Results

G◦(P) Delaunay G(P) L∞-Delaunay G△(P) G▽(P) Half-Θ6-Graphs G(P) Θ6-Graphs

  • Max. Matching µ(n)

Spanning Ratio t t = 2 1.593<t<1.998 t = 2 t = 2.61 µ(n) = ⌊n

2⌋

µ(n) = ⌈n−1

3 ⌉

µ(n) = ⌊n

2⌋

(even

  • Hamil. path)
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SLIDE 64

Known Results

G◦(P) Delaunay G(P) L∞-Delaunay G△(P) G▽(P) Half-Θ6-Graphs G(P) Θ6-Graphs

  • Max. Matching µ(n)

Spanning Ratio t t = 2 1.593<t<1.998 t = 2 t = 2.61 µ(n) = ⌊n

2⌋

µ(n) = ⌈n−1

3 ⌉

µ(n) = ⌊n

2⌋

(even

  • Hamil. path)

µ(n) ≥ ⌈n−1

3 ⌉

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SLIDE 65

Known Results

G◦(P) Delaunay G(P) L∞-Delaunay G△(P) G▽(P) Half-Θ6-Graphs G(P) Θ6-Graphs

  • Max. Matching µ(n)

Spanning Ratio t t = 2 1.593<t<1.998 t = 2 t = 2.61 µ(n) = ⌊n

2⌋

µ(n) = ⌈n−1

3 ⌉

µ(n) = ⌊n

2⌋

(even

  • Hamil. path)

µ(n) ≥ ⌈n−1

3 ⌉

µ(n) ≤ ⌊n

2⌋

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SLIDE 66

Known Results

G◦(P) Delaunay G(P) L∞-Delaunay G△(P) G▽(P) Half-Θ6-Graphs G(P) Θ6-Graphs

  • Max. Matching µ(n)

Spanning Ratio t t = 2 1.593<t<1.998 t = 2 t = 2.61 µ(n) = ⌊n

2⌋

µ(n) = ⌈n−1

3 ⌉

µ(n) = ⌊n

2⌋

(even

  • Hamil. path)

µ(n) ≥ ⌈n−1

3 ⌉

µ(n) ≤ ⌊n

2⌋

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SLIDE 67

The Tutte-Berge Formula

G = (V, E)

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SLIDE 68

The Tutte-Berge Formula

G = (V, E) S ⊆ V

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SLIDE 69

The Tutte-Berge Formula

G = (V, E) S ⊆ V

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SLIDE 70

The Tutte-Berge Formula

G = (V, E) S ⊆ V

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SLIDE 71

The Tutte-Berge Formula

G = (V, E) S ⊆ V comp(G \ S)

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SLIDE 72

The Tutte-Berge Formula

G = (V, E) S ⊆ V comp(G \ S)

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SLIDE 73

The Tutte-Berge Formula

G = (V, E) S ⊆ V comp(G \ S)

  • dd(G \ S)
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SLIDE 74

The Tutte-Berge Formula

G = (V, E) S ⊆ V comp(G \ S)

  • dd(G \ S)

[Tutte ’47] G has a (near-)perfect matching

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SLIDE 75

The Tutte-Berge Formula

G = (V, E) S ⊆ V comp(G \ S)

  • dd(G \ S)

[Tutte ’47]

⇔ odd(S) ≤ |S| for every S ⊆ V

G has a (near-)perfect matching

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SLIDE 76

The Tutte-Berge Formula

G = (V, E) S ⊆ V comp(G \ S)

  • dd(G \ S)

[Tutte ’47]

⇔ odd(S) ≤ |S| for every S ⊆ V

G has a (near-)perfect matching

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SLIDE 77

The Tutte-Berge Formula

G = (V, E) S ⊆ V comp(G \ S)

  • dd(G \ S)

[Tutte ’47]

⇔ odd(S) ≤ |S| for every S ⊆ V

[Berge ’57] G has a (near-)perfect matching

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SLIDE 78

The Tutte-Berge Formula

G = (V, E) S ⊆ V comp(G \ S)

  • dd(G \ S)

[Tutte ’47]

⇔ odd(S) ≤ |S| for every S ⊆ V

[Berge ’57] G has a (near-)perfect matching A maximum matching of G has size

slide-79
SLIDE 79

The Tutte-Berge Formula

G = (V, E) S ⊆ V comp(G \ S)

  • dd(G \ S)

[Tutte ’47]

⇔ odd(S) ≤ |S| for every S ⊆ V

[Berge ’57] G has a (near-)perfect matching A maximum matching of G has size

1 2(|V| − maxS⊆V (odd(G \ S) − |S|))

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SLIDE 80

The Tutte-Berge Formula

G = (V, E) S ⊆ V comp(G \ S)

  • dd(G \ S)

[Tutte ’47]

⇔ odd(S) ≤ |S| for every S ⊆ V

[Berge ’57] G has a (near-)perfect matching A maximum matching of G has size

1 2(|V| − maxS⊆V (odd(G \ S) − |S|))

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SLIDE 81

The Tutte-Berge Formula

G = (V, E) S ⊆ V comp(G \ S)

  • dd(G \ S)

[Tutte ’47]

⇔ odd(S) ≤ |S| for every S ⊆ V

[Berge ’57] G has a (near-)perfect matching A maximum matching of G has size

1 2(|V| − maxS⊆V (odd(G \ S) − |S|))

slide-82
SLIDE 82

The Tutte-Berge Formula

G = (V, E) S ⊆ V comp(G \ S)

  • dd(G \ S)

[Tutte ’47]

⇔ odd(S) ≤ |S| for every S ⊆ V

[Berge ’57] G has a (near-)perfect matching A maximum matching of G has size

1 2(|V| − maxS⊆V (odd(G \ S) − |S|))

⇒ there are maxS⊆V (odd(G \ S) − |S|) unmatched vtcs

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SLIDE 83

Upper Bound on Maximum Matching

degree of a face

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SLIDE 84

Upper Bound on Maximum Matching

degree of a face d = 3

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SLIDE 85

Upper Bound on Maximum Matching

degree of a face d = 3

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SLIDE 86

Upper Bound on Maximum Matching

degree of a face d = 3

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SLIDE 87

Upper Bound on Maximum Matching

degree of a face d = 3 d = 6

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SLIDE 88

Upper Bound on Maximum Matching

degree of a face d = 3 d = 6

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SLIDE 89

Upper Bound on Maximum Matching

degree of a face d = 3 d = 6

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SLIDE 90

Upper Bound on Maximum Matching

degree of a face d = 3 d = 6 d = 11

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SLIDE 91

Upper Bound on Maximum Matching

degree of a face d = 3 d = 6 d = 11 ∑ d≥3(d − 2)|F|△

deg d ≤ 2n − 4

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SLIDE 92

Upper Bound on Maximum Matching

degree of a face d = 3 d = 6 d = 11 G△ ∑ d≥3(d − 2)|F|△

deg d ≤ 2n − 4

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SLIDE 93

Upper Bound on Maximum Matching

degree of a face d = 3 d = 6 d = 11 G△ ∑ d≥3(d − 2)|F|△

deg d ≤ 2n − 4

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SLIDE 94

Upper Bound on Maximum Matching

degree of a face d = 3 d = 6 d = 11 G△ ∑ d≥3(d − 2)|F|△

deg d ≤ 2n − 4

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SLIDE 95

Upper Bound on Maximum Matching

degree of a face d = 3 d = 6 d = 11 G△ ∑ d≥3(d − 2)|F|△

deg d ≤ 2n − 4

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SLIDE 96

Upper Bound on Maximum Matching

degree of a face d = 3 d = 6 d = 11 G△ ∑ d≥3(d − 2)|F|△

deg d ≤ 2n − 4

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SLIDE 97

Upper Bound on Maximum Matching

degree of a face d = 3 d = 6 d = 11 G△ A ∑ d≥3(d − 2)|F|△

deg d ≤ 2n − 4

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SLIDE 98

Upper Bound on Maximum Matching

degree of a face d = 3 d = 6 d = 11 G△ A ∑ d≥3(d − 2)|F|△

deg d ≤ 2n − 4

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SLIDE 99

Upper Bound on Maximum Matching

degree of a face d = 3 d = 6 d = 11 G△ A ∑ d≥3(d − 2)|F|△

deg d ≤ 2n − 4

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SLIDE 100

Upper Bound on Maximum Matching

degree of a face d = 3 d = 6 d = 11 G△ A G△

A

∑ d≥3(d − 2)|F|△

deg d ≤ 2n − 4

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SLIDE 101

Upper Bound on Maximum Matching

degree of a face d = 3 d = 6 d = 11 G△ A G△

A

[Dillencourt ’90] G = (V, E) plane triangulation

⇒ ∀S ⊆ V :

∑ d≥3(d − 2)|F|△

deg d ≤ 2n − 4

slide-102
SLIDE 102

Upper Bound on Maximum Matching

degree of a face d = 3 d = 6 d = 11 G△ A G△

A

[Dillencourt ’90] G = (V, E) plane triangulation

⇒ ∀S ⊆ V :

∑ d≥3(d − 2)|F|△

deg d ≤ 2n − 4

slide-103
SLIDE 103

Upper Bound on Maximum Matching

degree of a face d = 3 d = 6 d = 11 G△ A G△

A

[Dillencourt ’90] G = (V, E) plane triangulation

⇒ ∀S ⊆ V : every face of G[S] contains ≤ 1 comp. of G \ S

∑ d≥3(d − 2)|F|△

deg d ≤ 2n − 4

slide-104
SLIDE 104

Upper Bound on Maximum Matching

degree of a face d = 3 d = 6 d = 11 G△ A G△

A

[Dillencourt ’90] G = (V, E) plane triangulation

⇒ ∀S ⊆ V : every face of G[S] contains ≤ 1 comp. of G \ S

G△

A [SA]

∑ d≥3(d − 2)|F|△

deg d ≤ 2n − 4

slide-105
SLIDE 105

Upper Bound on Maximum Matching

degree of a face d = 3 d = 6 d = 11 G△ A G△

A

[Dillencourt ’90] G = (V, E) plane triangulation

⇒ ∀S ⊆ V : every face of G[S] contains ≤ 1 comp. of G \ S

G△

A [SA]

∑ d≥3(d − 2)|F|△

deg d ≤ 2n − 4

slide-106
SLIDE 106

Upper Bound on Maximum Matching

degree of a face d = 3 d = 6 d = 11 G△ A G△

A

[Dillencourt ’90] G = (V, E) plane triangulation

⇒ ∀S ⊆ V : every face of G[S] contains ≤ 1 comp. of G \ S

G△

A [SA]

∑ d≥3(d − 2)|F|△

deg d ≤ 2n − 4

slide-107
SLIDE 107

Upper Bound on Maximum Matching

degree of a face d = 3 d = 6 d = 11 G△ A G△

A

[Dillencourt ’90] G = (V, E) plane triangulation

⇒ ∀S ⊆ V : every face of G[S] contains ≤ 1 comp. of G \ S

G△

A [SA]

∑ d≥3(d − 2)|F|△

deg d ≤ 2n − 4

slide-108
SLIDE 108

Upper Bound on Maximum Matching

degree of a face d = 3 d = 6 d = 11 G△ A G△

A

[Dillencourt ’90] G = (V, E) plane triangulation

⇒ ∀S ⊆ V : every face of G[S] contains ≤ 1 comp. of G \ S

G△

A [SA]

G△

A \ SA

∑ d≥3(d − 2)|F|△

deg d ≤ 2n − 4

slide-109
SLIDE 109

Upper Bound on Maximum Matching

G▽ degree of a face d = 3 d = 6 d = 11 G△ A G△

A

[Dillencourt ’90] G = (V, E) plane triangulation

⇒ ∀S ⊆ V : every face of G[S] contains ≤ 1 comp. of G \ S

G△

A [SA]

G△

A \ SA

∑ d≥3(d − 2)|F|△

deg d ≤ 2n − 4

slide-110
SLIDE 110

Upper Bound on Maximum Matching

G▽ degree of a face d = 3 d = 6 d = 11 G△ A G△

A

[Dillencourt ’90] G = (V, E) plane triangulation

⇒ ∀S ⊆ V : every face of G[S] contains ≤ 1 comp. of G \ S

G△

A [SA]

G▽

A

G△

A \ SA

∑ d≥3(d − 2)|F|△

deg d ≤ 2n − 4

slide-111
SLIDE 111

Upper Bound on Maximum Matching

G▽ degree of a face d = 3 d = 6 d = 11 G△ A G△

A

[Dillencourt ’90] G = (V, E) plane triangulation

⇒ ∀S ⊆ V : every face of G[S] contains ≤ 1 comp. of G \ S

G△

A [SA]

G▽

A

G△

A \ SA

G▽

A [SA]

∑ d≥3(d − 2)|F|△

deg d ≤ 2n − 4

slide-112
SLIDE 112

Upper Bound on Maximum Matching

G▽ degree of a face d = 3 d = 6 d = 11 G△ A G△

A

[Dillencourt ’90] G = (V, E) plane triangulation

⇒ ∀S ⊆ V : every face of G[S] contains ≤ 1 comp. of G \ S

G△

A [SA]

G▽

A

G△

A \ SA

G▽

A [SA]

G▽

A \ SA

∑ d≥3(d − 2)|F|△

deg d ≤ 2n − 4

slide-113
SLIDE 113

Upper Bound on Maximum Matching

G▽ degree of a face d = 3 d = 6 d = 11 G△ A G△

A

[Dillencourt ’90] G = (V, E) plane triangulation

⇒ ∀S ⊆ V : every face of G[S] contains ≤ 1 comp. of G \ S

G△

A [SA]

G▽

A

G G△

A \ SA

G▽

A [SA]

G▽

A \ SA

∑ d≥3(d − 2)|F|△

deg d ≤ 2n − 4

slide-114
SLIDE 114

Upper Bound on Maximum Matching

G▽ degree of a face d = 3 d = 6 d = 11 G△ A G△

A

[Dillencourt ’90] G = (V, E) plane triangulation

⇒ ∀S ⊆ V : every face of G[S] contains ≤ 1 comp. of G \ S

G△

A [SA]

G▽

A

G G△

A \ SA

G▽

A [SA]

G▽

A \ SA

S ∑ d≥3(d − 2)|F|△

deg d ≤ 2n − 4

slide-115
SLIDE 115

Upper Bound on Maximum Matching

G▽ degree of a face d = 3 d = 6 d = 11 G△ A G△

A

[Dillencourt ’90] G = (V, E) plane triangulation

⇒ ∀S ⊆ V : every face of G[S] contains ≤ 1 comp. of G \ S

G△

A [SA]

G▽

A

G G△

A \ SA

G▽

A [SA]

G▽

A \ SA

S comp(G \ S) ∑ d≥3(d − 2)|F|△

deg d ≤ 2n − 4

slide-116
SLIDE 116

Upper Bound on Maximum Matching

G▽ degree of a face d = 3 d = 6 d = 11 G△ A G△

A

[Dillencourt ’90] G = (V, E) plane triangulation

⇒ ∀S ⊆ V : every face of G[S] contains ≤ 1 comp. of G \ S

G△

A [SA]

G▽

A

G G△

A \ SA

G▽

A [SA]

G▽

A \ SA

S comp(G \ S)

|Q|=comp(G \ S)

∑ d≥3(d − 2)|F|△

deg d ≤ 2n − 4

slide-117
SLIDE 117

Upper Bound on Maximum Matching

G▽ degree of a face d = 3 d = 6 d = 11 G△ A G△

A

[Dillencourt ’90] G = (V, E) plane triangulation

⇒ ∀S ⊆ V : every face of G[S] contains ≤ 1 comp. of G \ S

G△

A [SA]

G▽

A

G G△

A \ SA

G▽

A [SA]

G▽

A \ SA

S comp(G \ S)

|Q|=comp(G \ S)

∑ d≥3(d − 2)|F|△

deg d ≤ 2n − 4

slide-118
SLIDE 118

Upper Bound on Maximum Matching

G▽ degree of a face d = 3 d = 6 d = 11 G△ A G△

A

[Dillencourt ’90] G = (V, E) plane triangulation

⇒ ∀S ⊆ V : every face of G[S] contains ≤ 1 comp. of G \ S

G△

A [SA]

G▽

A

G G△

A \ SA

G▽

A [SA]

G▽

A \ SA

S comp(G \ S)

|Q|=comp(G \ S)

q ∈ Q in deg-3 face of G△

A [SA]

⇒ q in deg-4+ face of G▽

A [SA]

∑ d≥3(d − 2)|F|△

deg d ≤ 2n − 4

slide-119
SLIDE 119

Upper Bound on Maximum Matching

G▽ degree of a face d = 3 d = 6 d = 11 G△ A G△

A

[Dillencourt ’90] G = (V, E) plane triangulation

⇒ ∀S ⊆ V : every face of G[S] contains ≤ 1 comp. of G \ S

G△

A [SA]

G▽

A

G G△

A \ SA

G▽

A [SA]

G▽

A \ SA

S comp(G \ S)

|Q|=comp(G \ S)

q ∈ Q in deg-3 face of G△

A [SA]

⇒ q in deg-4+ face of G▽

A [SA]

∑ d≥3(d − 2)|F|△

deg d ≤ 2n − 4

|F|△

deg 3 ≤ |F|▽ deg 4+

slide-120
SLIDE 120

Upper Bound on Maximum Matching

G▽ degree of a face d = 3 d = 6 d = 11 G△ A G△

A

[Dillencourt ’90] G = (V, E) plane triangulation

⇒ ∀S ⊆ V : every face of G[S] contains ≤ 1 comp. of G \ S

G△

A [SA]

G▽

A

G G△

A \ SA

G▽

A [SA]

G▽

A \ SA

S comp(G \ S)

|Q|=comp(G \ S)

q ∈ Q in deg-3 face of G△

A [SA]

⇒ q in deg-4+ face of G▽

A [SA]

comp(G \ S) − |S| = |Q| − |S| ≤ n+16

7

∑ d≥3(d − 2)|F|△

deg d ≤ 2n − 4

|F|△

deg 3 ≤ |F|▽ deg 4+

slide-121
SLIDE 121

Upper Bound on Maximum Matching

G▽ degree of a face d = 3 d = 6 d = 11 G△ A G△

A

[Dillencourt ’90] G = (V, E) plane triangulation

⇒ ∀S ⊆ V : every face of G[S] contains ≤ 1 comp. of G \ S

G△

A [SA]

G▽

A

G G△

A \ SA

G▽

A [SA]

G▽

A \ SA

S comp(G \ S)

|Q|=comp(G \ S)

q ∈ Q in deg-3 face of G△

A [SA]

⇒ q in deg-4+ face of G▽

A [SA]

µ(G)= 1

2(|V|− maxS⊆V (odd(G \ S)−|S|))

[Tutte-Berge] comp(G \ S) − |S| = |Q| − |S| ≤ n+16

7

∑ d≥3(d − 2)|F|△

deg d ≤ 2n − 4

|F|△

deg 3 ≤ |F|▽ deg 4+

slide-122
SLIDE 122

Upper Bound on Maximum Matching

G▽ degree of a face d = 3 d = 6 d = 11 G△ A G△

A

[Dillencourt ’90] G = (V, E) plane triangulation

⇒ ∀S ⊆ V : every face of G[S] contains ≤ 1 comp. of G \ S

G△

A [SA]

G▽

A

G G△

A \ SA

G▽

A [SA]

G▽

A \ SA

S comp(G \ S)

|Q|=comp(G \ S)

q ∈ Q in deg-3 face of G△

A [SA]

⇒ q in deg-4+ face of G▽

A [SA]

µ(G)= 1

2(|V|− maxS⊆V (odd(G \ S)−|S|))

[Tutte-Berge] comp(G \ S) − |S| = |Q| − |S| ≤ n+16

7

⇒ µ(n) ≥ 3n−8

7

∑ d≥3(d − 2)|F|△

deg d ≤ 2n − 4

|F|△

deg 3 ≤ |F|▽ deg 4+

slide-123
SLIDE 123

Known Results

G◦(P) Delaunay G(P) L∞-Delaunay G△(P) G▽(P) Half-Θ6-Graphs G(P) Θ6-Graphs

  • Max. Matching µ(n)

Spanning Ratio t t = 2 1.593<t<1.998 t = 2 t = 2.61 µ(n) = ⌊n

2⌋

µ(n) = ⌈n−1

3 ⌉

µ(n) = ⌊n

2⌋

(even

  • Hamil. path)

µ(n) ≥ ⌈n−1

3 ⌉

µ(n) ≤ ⌊n

2⌋

slide-124
SLIDE 124

Known Results

G◦(P) Delaunay G(P) L∞-Delaunay G△(P) G▽(P) Half-Θ6-Graphs G(P) Θ6-Graphs

  • Max. Matching µ(n)

Spanning Ratio t t = 2 1.593<t<1.998 t = 2 t = 2.61 µ(n) = ⌊n

2⌋

µ(n) = ⌈n−1

3 ⌉

µ(n) = ⌊n

2⌋

(even

  • Hamil. path)

µ(n) ≤ ⌊n

2⌋

µ(n) ≥ 3n−8

7

slide-125
SLIDE 125

Blocking Sets

G△(P)

slide-126
SLIDE 126

Blocking Sets

G△(P) Blocking Set Q:

slide-127
SLIDE 127

Blocking Sets

G△(P) Blocking Set Q: ∀(p, q) ∈ G△(P) : ∃r ∈ Q : r ∈ △(p, q)

slide-128
SLIDE 128

Blocking Sets

G△(P) Blocking Set Q: ∀(p, q) ∈ G△(P) : ∃r ∈ Q : r ∈ △(p, q)

slide-129
SLIDE 129

Blocking Sets

G△(P) Blocking Set Q: ∀(p, q) ∈ G△(P) : ∃r ∈ Q : r ∈ △(p, q)

slide-130
SLIDE 130

Blocking Sets

G△(P) Blocking Set Q: ∀(p, q) ∈ G△(P) : ∃r ∈ Q : r ∈ △(p, q)

slide-131
SLIDE 131

Blocking Sets

G△(P) Blocking Set Q: ∀(p, q) ∈ G△(P) : ∃r ∈ Q : r ∈ △(p, q)

slide-132
SLIDE 132

Blocking Sets

G△(P) Blocking Set Q: ∀(p, q) ∈ G△(P) : ∃r ∈ Q : r ∈ △(p, q)

slide-133
SLIDE 133

Blocking Sets

G△(P) Blocking Set Q: ∀(p, q) ∈ G△(P) : ∃r ∈ Q : r ∈ △(p, q)

slide-134
SLIDE 134

Blocking Sets

G△(P) Blocking Set Q: ∀(p, q) ∈ G△(P) : ∃r ∈ Q : r ∈ △(p, q)

slide-135
SLIDE 135

Blocking Sets

G△(P) Blocking Set Q: ∀(p, q) ∈ G△(P) : ∃r ∈ Q : r ∈ △(p, q)

slide-136
SLIDE 136

Blocking Sets

G△(P) Blocking Set Q:

∀(p, q) ∈ G△(P ∪ Q) : q ∈ Q

  • r:

∀(p, q) ∈ G△(P) : ∃r ∈ Q : r ∈ △(p, q)

slide-137
SLIDE 137

Known Results

G◦(P) Delaunay G(P) L∞-Delaunay G△(P) G▽(P) Half-Θ6-Graphs G(P) Θ6-Graphs

  • Max. Matching µ(n)

Spanning Ratio t t = 2 1.593<t<1.998 t = 2 t = 2.61 µ(n) = ⌊n

2⌋

µ(n) = ⌈n−1

3 ⌉

µ(n) = ⌊n

2⌋

(even

  • Hamil. path)

µ(n) ≤ ⌊n

2⌋

µ(n) ≥ 3n−8

7

slide-138
SLIDE 138

Known Results

G◦(P) Delaunay G(P) L∞-Delaunay G△(P) G▽(P) Half-Θ6-Graphs G(P) Θ6-Graphs

  • Max. Matching µ(n)

Spanning Ratio t t = 2 1.593<t<1.998 t = 2 t = 2.61 µ(n) = ⌊n

2⌋

µ(n) = ⌈n−1

3 ⌉

µ(n) = ⌊n

2⌋

(even

  • Hamil. path)
  • Min. Blocking Set

β(n) µ(n) ≤ ⌊n

2⌋

µ(n) ≥ 3n−8

7

slide-139
SLIDE 139

Known Results

G◦(P) Delaunay G(P) L∞-Delaunay G△(P) G▽(P) Half-Θ6-Graphs G(P) Θ6-Graphs

  • Max. Matching µ(n)

Spanning Ratio t t = 2 1.593<t<1.998 t = 2 t = 2.61 µ(n) = ⌊n

2⌋

µ(n) = ⌈n−1

3 ⌉

µ(n) = ⌊n

2⌋

(even

  • Hamil. path)
  • Min. Blocking Set

β(n) β(n) ≥ n − 1 µ(n) ≤ ⌊n

2⌋

µ(n) ≥ 3n−8

7

slide-140
SLIDE 140

Known Results

G◦(P) Delaunay G(P) L∞-Delaunay G△(P) G▽(P) Half-Θ6-Graphs G(P) Θ6-Graphs

  • Max. Matching µ(n)

Spanning Ratio t t = 2 1.593<t<1.998 t = 2 t = 2.61 µ(n) = ⌊n

2⌋

µ(n) = ⌈n−1

3 ⌉

µ(n) = ⌊n

2⌋

(even

  • Hamil. path)
  • Min. Blocking Set

β(n) β(n) ≥ n − 1 β(n) ≤ 3n

2

µ(n) ≤ ⌊n

2⌋

µ(n) ≥ 3n−8

7

slide-141
SLIDE 141

Known Results

G◦(P) Delaunay G(P) L∞-Delaunay G△(P) G▽(P) Half-Θ6-Graphs G(P) Θ6-Graphs

  • Max. Matching µ(n)

Spanning Ratio t t = 2 1.593<t<1.998 t = 2 t = 2.61 µ(n) = ⌊n

2⌋

µ(n) = ⌈n−1

3 ⌉

µ(n) = ⌊n

2⌋

(even

  • Hamil. path)
  • Min. Blocking Set

β(n) β(n) ≥ n − 1 β(n) ≤ 3n

2

β(n) ≥ 5n

4 −o(n)

µ(n) ≤ ⌊n

2⌋

µ(n) ≥ 3n−8

7

slide-142
SLIDE 142

Known Results

G◦(P) Delaunay G(P) L∞-Delaunay G△(P) G▽(P) Half-Θ6-Graphs G(P) Θ6-Graphs

  • Max. Matching µ(n)

Spanning Ratio t t = 2 1.593<t<1.998 t = 2 t = 2.61 µ(n) = ⌊n

2⌋

µ(n) = ⌈n−1

3 ⌉

µ(n) = ⌊n

2⌋

(even

  • Hamil. path)
  • Min. Blocking Set

β(n) β(n) ≥ n − 1 β(n) ≤ 3n

2

β(n) ≥ 5n

4 −o(n)

β(n) ≤ 2n−o(n) µ(n) ≤ ⌊n

2⌋

µ(n) ≥ 3n−8

7

slide-143
SLIDE 143

Known Results

G◦(P) Delaunay G(P) L∞-Delaunay G△(P) G▽(P) Half-Θ6-Graphs G(P) Θ6-Graphs

  • Max. Matching µ(n)

Spanning Ratio t t = 2 1.593<t<1.998 t = 2 t = 2.61 µ(n) = ⌊n

2⌋

µ(n) = ⌈n−1

3 ⌉

µ(n) = ⌊n

2⌋

(even

  • Hamil. path)
  • Min. Blocking Set

β(n) β(n) ≥ n − 1 β(n) ≤ 3n

2

β(n) = ⌈n−1

2 ⌉

β(n) ≥ 5n

4 −o(n)

β(n) ≤ 2n−o(n) µ(n) ≤ ⌊n

2⌋

µ(n) ≥ 3n−8

7

slide-144
SLIDE 144

Known Results

G◦(P) Delaunay G(P) L∞-Delaunay G△(P) G▽(P) Half-Θ6-Graphs G(P) Θ6-Graphs

  • Max. Matching µ(n)

Spanning Ratio t t = 2 1.593<t<1.998 t = 2 t = 2.61 µ(n) = ⌊n

2⌋

µ(n) = ⌈n−1

3 ⌉

µ(n) = ⌊n

2⌋

(even

  • Hamil. path)
  • Min. Blocking Set

β(n) β(n) ≥ n − 1 β(n) ≤ 3n

2

β(n) = ⌈n−1

2 ⌉

β(n) ≥ ⌈n−1

2 ⌉

β(n) ≥ 5n

4 −o(n)

β(n) ≤ 2n−o(n) µ(n) ≤ ⌊n

2⌋

µ(n) ≥ 3n−8

7

slide-145
SLIDE 145

Known Results

G◦(P) Delaunay G(P) L∞-Delaunay G△(P) G▽(P) Half-Θ6-Graphs G(P) Θ6-Graphs

  • Max. Matching µ(n)

Spanning Ratio t t = 2 1.593<t<1.998 t = 2 t = 2.61 µ(n) = ⌊n

2⌋

µ(n) = ⌈n−1

3 ⌉

µ(n) = ⌊n

2⌋

(even

  • Hamil. path)
  • Min. Blocking Set

β(n) β(n) ≥ n − 1 β(n) ≤ 3n

2

β(n) = ⌈n−1

2 ⌉

β(n) ≥ ⌈n−1

2 ⌉

β(n) ≥ 5n

4 −o(n)

β(n) ≤ 2n−o(n) µ(n) ≤ ⌊n

2⌋

µ(n) ≥ 3n−8

7

β(n) ≤ n − 1

slide-146
SLIDE 146

Relationship Blocking Set & Matching

G(P)

slide-147
SLIDE 147

Relationship Blocking Set & Matching

G(P) S

slide-148
SLIDE 148

Relationship Blocking Set & Matching

G(P) S comp(G(P) \ S)

slide-149
SLIDE 149

Relationship Blocking Set & Matching

G(P) S comp(G(P) \ S)

|Q| = comp(G(P) \ S)

slide-150
SLIDE 150

Relationship Blocking Set & Matching

G(P) S comp(G(P) \ S)

|Q| = comp(G(P) \ S)

slide-151
SLIDE 151

Relationship Blocking Set & Matching

G(P) S comp(G(P) \ S)

|Q| = comp(G(P) \ S)

G(Q)

slide-152
SLIDE 152

Relationship Blocking Set & Matching

G(P) S comp(G(P) \ S)

|Q| = comp(G(P) \ S)

G(Q)

p q

slide-153
SLIDE 153

Relationship Blocking Set & Matching

G(P) S comp(G(P) \ S)

|Q| = comp(G(P) \ S)

G(Q)

p q

△(p, q)

slide-154
SLIDE 154

Relationship Blocking Set & Matching

G(P) S comp(G(P) \ S)

|Q| = comp(G(P) \ S)

G(Q)

p q

△(p, q)

slide-155
SLIDE 155

Relationship Blocking Set & Matching

G(P) S comp(G(P) \ S)

|Q| = comp(G(P) \ S)

G(Q)

p q

△(p, q)

∃p, . . . , q s.t.

defining △s lie inside △(p, q) [Babu et al. ’14]

slide-156
SLIDE 156

Relationship Blocking Set & Matching

G(P) S comp(G(P) \ S)

|Q| = comp(G(P) \ S)

G(Q)

p q

△(p, q)

∃p, . . . , q s.t.

defining △s lie inside △(p, q) [Babu et al. ’14]

slide-157
SLIDE 157

Relationship Blocking Set & Matching

G(P) S comp(G(P) \ S)

|Q| = comp(G(P) \ S)

G(Q)

p q

△(p, q)

∃p, . . . , q s.t.

defining △s lie inside △(p, q) [Babu et al. ’14]

slide-158
SLIDE 158

Relationship Blocking Set & Matching

G(P) S comp(G(P) \ S)

|Q| = comp(G(P) \ S)

G(Q)

p q

△(p, q)

r

∃p, . . . , q s.t.

defining △s lie inside △(p, q) [Babu et al. ’14]

slide-159
SLIDE 159

Relationship Blocking Set & Matching

G(P) S comp(G(P) \ S)

|Q| = comp(G(P) \ S)

G(Q)

p q

△(p, q)

r

p, q in different comp. of G(P) \ S ⇒ r ∈ S

∃p, . . . , q s.t.

defining △s lie inside △(p, q) [Babu et al. ’14]

slide-160
SLIDE 160

Relationship Blocking Set & Matching

G(P) S comp(G(P) \ S)

|Q| = comp(G(P) \ S)

G(Q)

p q

△(p, q)

r

p, q in different comp. of G(P) \ S ⇒ r ∈ S

⇒ ∀(p, q) ∈ G(Q) : ∃r ∈ S : r ∈ △(p, q) ∃p, . . . , q s.t.

defining △s lie inside △(p, q) [Babu et al. ’14]

slide-161
SLIDE 161

Relationship Blocking Set & Matching

G(P) S comp(G(P) \ S)

|Q| = comp(G(P) \ S)

G(Q)

p q

△(p, q)

r

p, q in different comp. of G(P) \ S ⇒ r ∈ S

⇒ ∀(p, q) ∈ G(Q) : ∃r ∈ S : r ∈ △(p, q) ⇒ S is blocking set of Q ∃p, . . . , q s.t.

defining △s lie inside △(p, q) [Babu et al. ’14]

slide-162
SLIDE 162

Relationship Blocking Set & Matching

G(P) S comp(G(P) \ S)

|Q| = comp(G(P) \ S)

G(Q)

p q

△(p, q)

r

p, q in different comp. of G(P) \ S ⇒ r ∈ S

⇒ ∀(p, q) ∈ G(Q) : ∃r ∈ S : r ∈ △(p, q) ⇒ S is blocking set of Q ⇒ |S| ≥ β(|Q|) = β(comp(G(P) \ S)) ∃p, . . . , q s.t.

defining △s lie inside △(p, q) [Babu et al. ’14]

slide-163
SLIDE 163

Relationship Blocking Set & Matching

G(P) S comp(G(P) \ S)

|Q| = comp(G(P) \ S)

G(Q)

p q

△(p, q)

r

p, q in different comp. of G(P) \ S ⇒ r ∈ S

⇒ ∀(p, q) ∈ G(Q) : ∃r ∈ S : r ∈ △(p, q) ⇒ S is blocking set of Q ⇒ |S| ≥ β(|Q|) = β(comp(G(P) \ S)) ∃p, . . . , q s.t.

defining △s lie inside △(p, q) [Babu et al. ’14]

slide-164
SLIDE 164

Relationship Blocking Set & Matching

G(P) S comp(G(P) \ S)

|Q| = comp(G(P) \ S)

G(Q)

p q

△(p, q)

r

p, q in different comp. of G(P) \ S ⇒ r ∈ S

⇒ ∀(p, q) ∈ G(Q) : ∃r ∈ S : r ∈ △(p, q) ⇒ S is blocking set of Q ⇒ |S| ≥ β(|Q|) = β(comp(G(P) \ S)) ∃p, . . . , q s.t.

defining △s lie inside △(p, q) [Babu et al. ’14] µ(G)= 1

2(|V|− maxS⊆V (odd(G \ S)−|S|))

[Tutte-Berge]

slide-165
SLIDE 165

Relationship Blocking Set & Matching

G(P) S comp(G(P) \ S)

|Q| = comp(G(P) \ S)

G(Q)

p q

△(p, q)

r

p, q in different comp. of G(P) \ S ⇒ r ∈ S

⇒ ∀(p, q) ∈ G(Q) : ∃r ∈ S : r ∈ △(p, q) ⇒ S is blocking set of Q ⇒ |S| ≥ β(|Q|) = β(comp(G(P) \ S)) ∃p, . . . , q s.t.

defining △s lie inside △(p, q) [Babu et al. ’14]

⇒ µ(n) ≥ n−(|Q|−β(|Q|))

2

≥ n−(n−β(n))

2

= β(n)

2

µ(G)= 1

2(|V|− maxS⊆V (odd(G \ S)−|S|))

[Tutte-Berge]

slide-166
SLIDE 166

Relationship Blocking Set & Matching

G(P) S comp(G(P) \ S)

|Q| = comp(G(P) \ S)

G(Q)

p q

△(p, q)

r

p, q in different comp. of G(P) \ S ⇒ r ∈ S

⇒ ∀(p, q) ∈ G(Q) : ∃r ∈ S : r ∈ △(p, q) ⇒ S is blocking set of Q ⇒ |S| ≥ β(|Q|) = β(comp(G(P) \ S)) ∃p, . . . , q s.t.

defining △s lie inside △(p, q) [Babu et al. ’14]

⇒ µ(n) ≥ n−(|Q|−β(|Q|))

2

≥ n−(n−β(n))

2

= β(n)

2

µ(G)= 1

2(|V|− maxS⊆V (odd(G \ S)−|S|))

[Tutte-Berge]

slide-167
SLIDE 167

Relationship Blocking Set & Matching

G(P) S comp(G(P) \ S)

|Q| = comp(G(P) \ S)

G(Q)

p q

△(p, q)

r

p, q in different comp. of G(P) \ S ⇒ r ∈ S

⇒ ∀(p, q) ∈ G(Q) : ∃r ∈ S : r ∈ △(p, q) ⇒ S is blocking set of Q ⇒ |S| ≥ β(|Q|) = β(comp(G(P) \ S)) ∃p, . . . , q s.t.

defining △s lie inside △(p, q) [Babu et al. ’14]

⇒ µ(n) ≥ n−(|Q|−β(|Q|))

2

≥ n−(n−β(n))

2

= β(n)

2

µ(G)= 1

2(|V|− maxS⊆V (odd(G \ S)−|S|))

[Tutte-Berge]

slide-168
SLIDE 168

Relationship Blocking Set & Matching

G(P) S comp(G(P) \ S)

|Q| = comp(G(P) \ S)

G(Q)

p q

△(p, q)

r

p, q in different comp. of G(P) \ S ⇒ r ∈ S

⇒ ∀(p, q) ∈ G(Q) : ∃r ∈ S : r ∈ △(p, q) ⇒ S is blocking set of Q ⇒ |S| ≥ β(|Q|) = β(comp(G(P) \ S)) ∃p, . . . , q s.t.

defining △s lie inside △(p, q) [Babu et al. ’14]

⇒ µ(n) ≥ n−(|Q|−β(|Q|))

2

≥ n−(n−β(n))

2

= β(n)

2

µ(G)= 1

2(|V|− maxS⊆V (odd(G \ S)−|S|))

[Tutte-Berge]

slide-169
SLIDE 169

G◦(P) Delaunay G(P) L∞-Delaunay G△(P) G▽(P) Half-Θ6-Graphs G(P) Θ6-Graphs

  • Max. Matching µ(n)

Spanning Ratio t t = 2 1.593<t<1.998 t = 2 t = 2.61 µ(n) = ⌊n

2⌋

µ(n) = ⌈n−1

3 ⌉

µ(n) = ⌊n

2⌋

(even

  • Hamil. path)
  • Min. Blocking Set

β(n) β(n) ≥ n − 1 β(n) ≤ 3n

2

β(n) = ⌈n−1

2 ⌉

β(n) ≥ ⌈n−1

2 ⌉

β(n) ≥ 5n

4 −o(n)

β(n) ≤ 2n−o(n) µ(n) ≤ ⌊n

2⌋

µ(n) ≥ 3n−8

7

β(n) ≤ n − 1

slide-170
SLIDE 170

G◦(P) Delaunay G(P) L∞-Delaunay G△(P) G▽(P) Half-Θ6-Graphs G(P) Θ6-Graphs

  • Max. Matching µ(n)

Spanning Ratio t t = 2 1.593<t<1.998 t = 2 t = 2.61 µ(n) = ⌊n

2⌋

µ(n) = ⌈n−1

3 ⌉

µ(n) = ⌊n

2⌋

(even

  • Hamil. path)
  • Min. Blocking Set

β(n) β(n) ≥ n − 1 β(n) ≤ 3n

2

β(n) = ⌈n−1

2 ⌉

β(n) ≥ ⌈n−1

2 ⌉

β(n) ≥ 5n

4 −o(n)

β(n) ≤ 2n−o(n) µ(n) ≤ ⌊n

2⌋

µ(n) ≥ 3n−8

7

β(n) ≤ n − 1 µ(n) ≥ β(n)

2

slide-171
SLIDE 171

G◦(P) Delaunay G(P) L∞-Delaunay G△(P) G▽(P) Half-Θ6-Graphs G(P) Θ6-Graphs

  • Max. Matching µ(n)

Spanning Ratio t t = 2 1.593<t<1.998 t = 2 t = 2.61 µ(n) = ⌊n

2⌋

µ(n) = ⌈n−1

3 ⌉

µ(n) = ⌊n

2⌋

(even

  • Hamil. path)
  • Min. Blocking Set

β(n) β(n) ≥ n − 1 β(n) ≤ 3n

2

β(n) = ⌈n−1

2 ⌉

β(n) ≥ ⌈n−1

2 ⌉

β(n) ≥ 5n

4 −o(n)

β(n) ≤ 2n−o(n) µ(n) ≤ ⌊n

2⌋

µ(n) ≥ 3n−8

7

β(n) ≤ n − 1 µ(n) ≥ β(n)

2

β(n) ≥ 3n−8

4

slide-172
SLIDE 172

G◦(P) Delaunay G(P) L∞-Delaunay G△(P) G▽(P) Half-Θ6-Graphs G(P) Θ6-Graphs

  • Max. Matching µ(n)

Spanning Ratio t t = 2 1.593<t<1.998 t = 2 t = 2.61 µ(n) = ⌊n

2⌋

µ(n) = ⌈n−1

3 ⌉

µ(n) = ⌊n

2⌋

(even

  • Hamil. path)
  • Min. Blocking Set

β(n) β(n) ≥ n − 1 β(n) ≤ 3n

2

β(n) = ⌈n−1

2 ⌉

β(n) ≥ ⌈n−1

2 ⌉

β(n) ≥ 5n

4 −o(n)

β(n) ≤ 2n−o(n) µ(n) ≤ ⌊n

2⌋

µ(n) ≥ 3n−8

7

β(n) ≤ n − 1 µ(n) ≥ β(n)

2

β(n) ≥ 3n−8

4

µ(n) ≥ ⌊ n

2 ⌋?

slide-173
SLIDE 173

G◦(P) Delaunay G(P) L∞-Delaunay G△(P) G▽(P) Half-Θ6-Graphs G(P) Θ6-Graphs

  • Max. Matching µ(n)

Spanning Ratio t t = 2 1.593<t<1.998 t = 2 t = 2.61 µ(n) = ⌊n

2⌋

µ(n) = ⌈n−1

3 ⌉

µ(n) = ⌊n

2⌋

(even

  • Hamil. path)
  • Min. Blocking Set

β(n) β(n) ≥ n − 1 β(n) ≤ 3n

2

β(n) = ⌈n−1

2 ⌉

β(n) ≥ ⌈n−1

2 ⌉

β(n) ≥ 5n

4 −o(n)

β(n) ≤ 2n−o(n) µ(n) ≤ ⌊n

2⌋

µ(n) ≥ 3n−8

7

β(n) ≤ n − 1 µ(n) ≥ β(n)

2

β(n) ≥ 3n−8

4

µ(n) ≥ ⌊ n

2 ⌋?

β(n) ≥ n − 1?

slide-174
SLIDE 174

G◦(P) Delaunay G(P) L∞-Delaunay G△(P) G▽(P) Half-Θ6-Graphs G(P) Θ6-Graphs

  • Max. Matching µ(n)

Spanning Ratio t t = 2 1.593<t<1.998 t = 2 t = 2.61 µ(n) = ⌊n

2⌋

µ(n) = ⌈n−1

3 ⌉

µ(n) = ⌊n

2⌋

(even

  • Hamil. path)
  • Min. Blocking Set

β(n) β(n) ≥ n − 1 β(n) ≤ 3n

2

β(n) = ⌈n−1

2 ⌉

β(n) ≥ ⌈n−1

2 ⌉

β(n) ≥ 5n

4 −o(n)

β(n) ≤ 2n−o(n) µ(n) ≤ ⌊n

2⌋

µ(n) ≥ 3n−8

7

β(n) ≤ n − 1 µ(n) ≥ β(n)

2

β(n) ≥ 3n−8

4

µ(n) ≥ ⌊ n

2 ⌋?

β(n) ≥ n − 1?

⇔