Edge Guarding Plane Graphs March 17, 2020 Paul Jungeblut, Torsten - - PowerPoint PPT Presentation

▶
edge guarding plane graphs
SMART_READER_LITE
LIVE PREVIEW

Edge Guarding Plane Graphs March 17, 2020 Paul Jungeblut, Torsten - - PowerPoint PPT Presentation

Oct 06, 2022 253 likes •667 views

Edge Guarding Plane Graphs March 17, 2020 Paul Jungeblut, Torsten Ueckerdt I NSTITUTE OF T HEORETICAL I NFORMATICS A LGORITHMICS G ROUP KIT The Research University in the Helmholtz Association www.kit.edu Edge Guarding G = ( V , E )

slide-1
SLIDE 1

Edge Guarding Plane Graphs

March 17, 2020 Paul Jungeblut, Torsten Ueckerdt

KIT – The Research University in the Helmholtz Association

INSTITUTE OF THEORETICAL INFORMATICS · ALGORITHMICS GROUP

www.kit.edu

slide-2
SLIDE 2

Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Edge Guarding

G = (V, E) plane graph. vw guards face f if at least one from {v, w} is on the boundary of f.

1

slide-3
SLIDE 3

Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Edge Guarding

G = (V, E) plane graph. vw guards face f if at least one from {v, w} is on the boundary of f.

1

slide-4
SLIDE 4

Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Edge Guarding

G = (V, E) plane graph. vw guards face f if at least one from {v, w} is on the boundary of f.

1

slide-5
SLIDE 5

Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Edge Guarding

G = (V, E) plane graph. vw guards face f if at least one from {v, w} is on the boundary of f.

1

slide-6
SLIDE 6

Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Edge Guarding

G = (V, E) plane graph. vw guards face f if at least one from {v, w} is on the boundary of f.

Question

For all n-vertex graphs of a planar graph class C: How many guards are sometimes necessary and always sufficient?

1

slide-7
SLIDE 7

Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Previous Results

Planar Triangulation Outerplanar

  • Max. Outerplanar

Lower Upper min

3n

8

  • ,

n

3 + α 9

2 4n−8

13

1 n

3

1 n

4

5 n

3

3 n

3

4 n

4

5

α: number of quadrilateral faces

n

3

1

1 Bose, Shermer, Toussaint, Zhu 1997 2 Biniaz, Bose, Ooms, Verdonschot 2019 3 Everett, Rivera-Campo 1997 4 Chv´

atal 1975

5 O’Rourke 1983 2

slide-8
SLIDE 8

Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Our Results

Stacked Triangulations Quadrangulations Lower Upper

2n

7

  • n−2

4

  • n

3

  • 2n−4

7

  • 2-Degenerate Quadrangulations

n−2

4

  • n

4

  • 3
slide-9
SLIDE 9

Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Our Results

Stacked Triangulations Quadrangulations Lower Upper

2n

7

  • n−2

4

  • n

3

  • 2n−4

7

  • 2-Degenerate Quadrangulations

n−2

4

  • n

4

  • Today!

3

slide-10
SLIDE 10

Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Definition: Stacked Triangulations

A triangle is a stacked triangulation. Let f be an inner face of a stacked triangulation: Adding a new vertex into f and subdividing it into three new faces gives a stacked triangulation.

4

slide-11
SLIDE 11

Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Definition: Stacked Triangulations

A triangle is a stacked triangulation. Let f be an inner face of a stacked triangulation: Adding a new vertex into f and subdividing it into three new faces gives a stacked triangulation.

4

slide-12
SLIDE 12

Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Definition: Stacked Triangulations

A triangle is a stacked triangulation. Let f be an inner face of a stacked triangulation: Adding a new vertex into f and subdividing it into three new faces gives a stacked triangulation.

4

slide-13
SLIDE 13

Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Definition: Stacked Triangulations

A triangle is a stacked triangulation. Let f be an inner face of a stacked triangulation: Adding a new vertex into f and subdividing it into three new faces gives a stacked triangulation.

4

slide-14
SLIDE 14

Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Definition: Stacked Triangulations

A triangle is a stacked triangulation. Let f be an inner face of a stacked triangulation: Adding a new vertex into f and subdividing it into three new faces gives a stacked triangulation.

4

slide-15
SLIDE 15

Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Definition: Stacked Triangulations

A triangle is a stacked triangulation. Let f be an inner face of a stacked triangulation: Adding a new vertex into f and subdividing it into three new faces gives a stacked triangulation.

Theorem [J. 2019]

For n-vertex stacked triangulations

2n

7

  • edge guards are

always sufficient.

4

slide-16
SLIDE 16

Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Induction via Vertex Deletion

Use induction on the number n of vertices:

  • 1. Create smaller graph G′
  • f size |G′| = |G| − k.
  • 2. Apply induction hypothesis on G′

to get edge guard set Γ ′.

  • 3. Reinsert old vertices.
  • 4. Use ℓ additional edges

to augment Γ ′ into Γ for G.

5

slide-17
SLIDE 17

Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Induction via Vertex Deletion

Use induction on the number n of vertices:

  • 1. Create smaller graph G′
  • f size |G′| = |G| − k.
  • 2. Apply induction hypothesis on G′

to get edge guard set Γ ′.

  • 3. Reinsert old vertices.
  • 4. Use ℓ additional edges

to augment Γ ′ into Γ for G.

5

slide-18
SLIDE 18

Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Induction via Vertex Deletion

Use induction on the number n of vertices:

  • 1. Create smaller graph G′
  • f size |G′| = |G| − k.
  • 2. Apply induction hypothesis on G′

to get edge guard set Γ ′.

  • 3. Reinsert old vertices.
  • 4. Use ℓ additional edges

to augment Γ ′ into Γ for G.

5

slide-19
SLIDE 19

Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Induction via Vertex Deletion

Use induction on the number n of vertices:

  • 1. Create smaller graph G′
  • f size |G′| = |G| − k.
  • 2. Apply induction hypothesis on G′

to get edge guard set Γ ′.

  • 3. Reinsert old vertices.
  • 4. Use ℓ additional edges

to augment Γ ′ into Γ for G.

5

slide-20
SLIDE 20

Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Induction via Vertex Deletion

Use induction on the number n of vertices:

  • 1. Create smaller graph G′
  • f size |G′| = |G| − k.
  • 2. Apply induction hypothesis on G′

to get edge guard set Γ ′.

  • 3. Reinsert old vertices.
  • 4. Use ℓ additional edges

to augment Γ ′ into Γ for G.

5

slide-21
SLIDE 21

Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Induction via Vertex Deletion

Use induction on the number n of vertices:

  • 1. Create smaller graph G′
  • f size |G′| = |G| − k.
  • 2. Apply induction hypothesis on G′

to get edge guard set Γ ′.

  • 3. Reinsert old vertices.
  • 4. Use ℓ additional edges

to augment Γ ′ into Γ for G.

ℓ

k ≤ 2 7 in all cases ⇒ edge guard set of size

2n

7

  • 5
slide-22
SLIDE 22

Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Induction via Vertex Deletion

Use induction on the number n of vertices:

  • 1. Create smaller graph G′
  • f size |G′| = |G| − k.
  • 2. Apply induction hypothesis on G′

to get edge guard set Γ ′.

  • 3. Reinsert old vertices.
  • 4. Use ℓ additional edges

to augment Γ ′ into Γ for G.

ℓ

k ≤ 2 7 in all cases ⇒ edge guard set of size

2n

7

  • Also applied successfully for 2-Degenerate Quadrangulations

ℓ

k ≤ 1 4

  • .

5

slide-23
SLIDE 23

Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Induction: Examples

stacked triangulation

6

slide-24
SLIDE 24

Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Induction: Examples

stacked triangulation Remove inner vertices (k = 6).

6

slide-25
SLIDE 25

Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Induction: Examples

stacked triangulation Remove inner vertices (k = 6). Apply induction.

6

slide-26
SLIDE 26

Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Induction: Examples

stacked triangulation Remove inner vertices (k = 6). Apply induction. Reinsert inner vertices.

6

slide-27
SLIDE 27

Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Induction: Examples

stacked triangulation Remove inner vertices (k = 6). Apply induction. Reinsert inner vertices. Add addtional edge (ℓ = 1), so ℓ

k = 1 6 ≤ 2 7.

6

slide-28
SLIDE 28

Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Induction: Examples

stacked triangulation Remove inner vertices (k = 6). Apply induction.

6

slide-29
SLIDE 29

Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Induction: Examples

stacked triangulation Remove inner vertices (k = 6). Apply induction. Reinsert inner vertices.

6

slide-30
SLIDE 30

Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Induction: Examples

stacked triangulation Remove inner vertices (k = 6). Apply induction. Reinsert inner vertices. Problem: Two edges are necessary for the remaining faces.

6

slide-31
SLIDE 31

Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Induction: Trick

somewhere in a stacked triangulation

Lemma

There is a minimum size edge guard set Γ with x, y ∈ V(Γ). x y

7

slide-32
SLIDE 32

Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Induction: Trick

somewhere in a stacked triangulation

Lemma

There is a minimum size edge guard set Γ with x, y ∈ V(Γ). x y

7

slide-33
SLIDE 33

Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Induction: Trick

somewhere in a stacked triangulation

Lemma

There is a minimum size edge guard set Γ with x, y ∈ V(Γ). x y

7

slide-34
SLIDE 34

Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Induction Example: Revisited

stacked triangulation Remove inner vertices (k = 6). Apply induction. Reinsert inner vertices. Problem: Two edges are necessary for the remaining faces.

8

slide-35
SLIDE 35

Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Induction Example: Revisited

stacked triangulation Remove inner vertices (k = 6).

8

slide-36
SLIDE 36

Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Induction Example: Revisited

stacked triangulation Remove inner vertices (k = 6). Add two new vertices (k = 6 k = 4).

8

slide-37
SLIDE 37

Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Induction Example: Revisited

stacked triangulation Remove inner vertices (k = 6). Add two new vertices (k = 6 k = 4). Apply lemma from last slide.

8

slide-38
SLIDE 38

Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Induction Example: Revisited

stacked triangulation Remove inner vertices (k = 6). Add two new vertices (k = 6 k = 4). Apply lemma from last slide. Reinsert old vertices.

8

slide-39
SLIDE 39

Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Induction Example: Revisited

stacked triangulation Remove inner vertices (k = 6). Add two new vertices (k = 6 k = 4). Apply lemma from last slide. Reinsert old vertices. One more edge suffices (ℓ = 1), so ℓ

k = 1 4 ≤ 2 7.

8

slide-40
SLIDE 40

Paul Jungeblut, Torsten Ueckerdt – EuroCG 2020, Edge Guarding Plane Graphs Institute of Theoretical Informatics Algorithmics Group

Open Problems

How many edge guards are always sufficient for general plane graphs? (4-connected) triangulations? quadrangulations?

Thank your for your attention.

9