36th European Workshop on Computational Geometry Disjoint - - PowerPoint PPT Presentation

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36th European Workshop on Computational Geometry Disjoint - - PowerPoint PPT Presentation

May 15, 2023 613 likes •1.29k views

36th European Workshop on Computational Geometry Disjoint tree-compatible plane perfect matchings Oswin Aichholzer 1 , Julia Obmann 1 , Pavel Pat ak 2 , Daniel Perz 1 , and Josef Tkadlec 2 1 Graz University of Technology, Austria 2 IST Austria,

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Julia Obmann Disjoint tree-compatible plane perfect matchings

1

Disjoint tree-compatible plane perfect matchings

Oswin Aichholzer1, Julia Obmann1, Pavel Pat´ ak2, Daniel Perz1, and Josef Tkadlec2

1 Graz University of Technology, Austria 2 IST Austria, Klosterneuburg, Austria

36th European Workshop on Computational Geometry

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Julia Obmann Disjoint tree-compatible plane perfect matchings

2 i

Compatibility of graphs

Setting: set S of 2n points in the plane in general position

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Julia Obmann Disjoint tree-compatible plane perfect matchings

2 ii

Compatibility of graphs

Setting: set S of 2n points in the plane in general position

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Julia Obmann Disjoint tree-compatible plane perfect matchings

2 iii

Compatibility of graphs

Setting: set S of 2n points in the plane in general position

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Julia Obmann Disjoint tree-compatible plane perfect matchings

2 iv

Compatibility of graphs

Setting: set S of 2n points in the plane in general position

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Julia Obmann Disjoint tree-compatible plane perfect matchings

2 v

Compatibility of graphs

Setting: set S of 2n points in the plane in general position compatible

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Julia Obmann Disjoint tree-compatible plane perfect matchings

2 vi

Compatibility of graphs

Setting: set S of 2n points in the plane in general position

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Julia Obmann Disjoint tree-compatible plane perfect matchings

2 vii

Compatibility of graphs

Setting: set S of 2n points in the plane in general position

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Julia Obmann Disjoint tree-compatible plane perfect matchings

2 viii

Compatibility of graphs

Setting: set S of 2n points in the plane in general position disjoint compatible

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Julia Obmann Disjoint tree-compatible plane perfect matchings

3 i

Compatibility of matchings

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Julia Obmann Disjoint tree-compatible plane perfect matchings

3 ii

Compatibility of matchings

  • compatibility graph:
  • vertices: all plane perfect matchings on S
  • edge (Mi, Mj) ⇐

⇒ Mi and Mj are compatible

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Julia Obmann Disjoint tree-compatible plane perfect matchings

3 iii

Compatibility of matchings

  • compatibility graph:
  • vertices: all plane perfect matchings on S
  • edge (Mi, Mj) ⇐

⇒ Mi and Mj are compatible

  • compatibility graph for matchings is connected

convex point set: [C. Hernando, F. Hurtado and M. Noy; 2002.] general point set: [M.E. Houle, F. Hurtado, M. Noy and

  • E. Rivera-Campo; 2005.]
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Julia Obmann Disjoint tree-compatible plane perfect matchings

3 iv

Compatibility of matchings

  • compatibility graph:
  • vertices: all plane perfect matchings on S
  • edge (Mi, Mj) ⇐

⇒ Mi and Mj are compatible

  • compatibility graph for matchings is connected

convex point set: [C. Hernando, F. Hurtado and M. Noy; 2002.] general point set: [M.E. Houle, F. Hurtado, M. Noy and

  • E. Rivera-Campo; 2005.]
  • diameter is O(log n) [ABDGHHKMRSSUW; 2009.]

and Ω(log n/ log log n) [A.Razen; 2008.]

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Julia Obmann Disjoint tree-compatible plane perfect matchings

4 i

Disjoint compatibility of matchings

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Julia Obmann Disjoint tree-compatible plane perfect matchings

4 ii

Disjoint compatibility of matchings

  • disjoint compatibility graph:
  • vertices: all plane perfect matchings on S
  • edge (Mi, Mj) ⇐

⇒ Mi, Mj are disjoint compatible

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Julia Obmann Disjoint tree-compatible plane perfect matchings

4 iii

Disjoint compatibility of matchings

  • disjoint compatibility graph:
  • vertices: all plane perfect matchings on S
  • edge (Mi, Mj) ⇐

⇒ Mi, Mj are disjoint compatible

  • disjoint compatibility graph for matchings on point sets
  • f 2n ≥ 6 points in convex position is disconnected

[O. Aichholzer, A. Asinowski and T. Miltzow; 2015.]

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Julia Obmann Disjoint tree-compatible plane perfect matchings

4 iv

Disjoint compatibility of matchings

  • disjoint compatibility graph:
  • vertices: all plane perfect matchings on S
  • edge (Mi, Mj) ⇐

⇒ Mi, Mj are disjoint compatible

  • disjoint compatibility graph for matchings on point sets
  • f 2n ≥ 6 points in convex position is disconnected

[O. Aichholzer, A. Asinowski and T. Miltzow; 2015.] Alternative way of defining compatibility?

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Julia Obmann Disjoint tree-compatible plane perfect matchings

5 i

Disjoint tree-compatibility of matchings

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Julia Obmann Disjoint tree-compatible plane perfect matchings

5 ii

Disjoint tree-compatibility of matchings

  • consider ’compatibility’ via

disjoint compatible plane spanning trees

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Julia Obmann Disjoint tree-compatible plane perfect matchings

5 iii

Disjoint tree-compatibility of matchings

  • consider ’compatibility’ via

disjoint compatible plane spanning trees

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Julia Obmann Disjoint tree-compatible plane perfect matchings

5 iv

Disjoint tree-compatibility of matchings

  • consider ’compatibility’ via

disjoint compatible plane spanning trees

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Julia Obmann Disjoint tree-compatible plane perfect matchings

5 v

Disjoint tree-compatibility of matchings

  • consider ’compatibility’ via

disjoint compatible plane spanning trees not compatible!

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Julia Obmann Disjoint tree-compatible plane perfect matchings

5 vi

Disjoint tree-compatibility of matchings

  • consider ’compatibility’ via

disjoint compatible plane spanning trees

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Julia Obmann Disjoint tree-compatible plane perfect matchings

5 vii

Disjoint tree-compatibility of matchings

  • consider ’compatibility’ via

disjoint compatible plane spanning trees

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Julia Obmann Disjoint tree-compatible plane perfect matchings

5 viii

Disjoint tree-compatibility of matchings

  • consider ’compatibility’ via

disjoint compatible plane spanning trees matchings are disjoint tree-compatible

(for short: tree-compatible)

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Julia Obmann Disjoint tree-compatible plane perfect matchings

5 ix

Disjoint tree-compatibility of matchings

  • consider ’compatibility’ via

disjoint compatible plane spanning trees

  • disjoint tree-compatibility graph G2n:
  • vertices: all plane perfect matchings on S
  • edge (Mi, Mj) ⇐

⇒ Mi, Mj disjoint tree-compatible

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Julia Obmann Disjoint tree-compatible plane perfect matchings

5 x

Disjoint tree-compatibility of matchings

  • consider ’compatibility’ via

disjoint compatible plane spanning trees

  • disjoint tree-compatibility graph G2n:
  • vertices: all plane perfect matchings on S
  • edge (Mi, Mj) ⇐

⇒ Mi, Mj disjoint tree-compatible ATTENTION: different from (disjoint) compatibility! disjoint tree-compatible compatible disjoint compatible disjoint tree-compatible

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Julia Obmann Disjoint tree-compatible plane perfect matchings

5 xi

Disjoint tree-compatibility of matchings

  • consider ’compatibility’ via

disjoint compatible plane spanning trees ATTENTION: different from (disjoint) compatibility! disjoint tree-compatible compatible disjoint compatible disjoint tree-compatible disjoint compatible NOT disjoint tree-compatible

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Julia Obmann Disjoint tree-compatible plane perfect matchings

6 i

G8

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Julia Obmann Disjoint tree-compatible plane perfect matchings

6 ii

G8

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Julia Obmann Disjoint tree-compatible plane perfect matchings

6 iii

G8

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Julia Obmann Disjoint tree-compatible plane perfect matchings

6 iv

G8

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Julia Obmann Disjoint tree-compatible plane perfect matchings

6 v

G8

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Julia Obmann Disjoint tree-compatible plane perfect matchings

6 vi

G8

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Julia Obmann Disjoint tree-compatible plane perfect matchings

6 vii

G8

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Julia Obmann Disjoint tree-compatible plane perfect matchings

6 viii

G8

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Julia Obmann Disjoint tree-compatible plane perfect matchings

6 ix

G8

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Julia Obmann Disjoint tree-compatible plane perfect matchings

7 i

Upper bound

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Julia Obmann Disjoint tree-compatible plane perfect matchings

7 ii

Upper bound

Theorem 1. For 2n ≥ 10, the graph G2n is connected and diam(G2n)≤ 5.

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Julia Obmann Disjoint tree-compatible plane perfect matchings

7 iii

Upper bound

Theorem 1. For 2n ≥ 10, the graph G2n is connected and diam(G2n)≤ 5. Proof Idea.

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Julia Obmann Disjoint tree-compatible plane perfect matchings

7 iv

Upper bound

Theorem 1. For 2n ≥ 10, the graph G2n is connected and diam(G2n)≤ 5. Proof Idea.

  • ”inside semicycles” can be (simultaneously) rotated in
  • ne step
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Julia Obmann Disjoint tree-compatible plane perfect matchings

7 v

Upper bound

Theorem 1. For 2n ≥ 10, the graph G2n is connected and diam(G2n)≤ 5. Proof Idea. X1 X2

  • ”inside semicycles” can be (simultaneously) rotated in
  • ne step
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Julia Obmann Disjoint tree-compatible plane perfect matchings

7 vi

Upper bound

Theorem 1. For 2n ≥ 10, the graph G2n is connected and diam(G2n)≤ 5. Proof Idea. X1 X2

  • ”inside semicycles” can be (simultaneously) rotated in
  • ne step
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Julia Obmann Disjoint tree-compatible plane perfect matchings

7 vii

Upper bound

Theorem 1. For 2n ≥ 10, the graph G2n is connected and diam(G2n)≤ 5. Proof Idea.

  • ”inside semicycles” can be (simultaneously) rotated in
  • ne step
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Julia Obmann Disjoint tree-compatible plane perfect matchings

7 viii

Upper bound

Theorem 1. For 2n ≥ 10, the graph G2n is connected and diam(G2n)≤ 5. Proof Idea.

  • large ”semiears” (≥ 12 vertices) can be rotated in at

most 3 steps

  • ”inside semicycles” can be (simultaneously) rotated in
  • ne step
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Julia Obmann Disjoint tree-compatible plane perfect matchings

7 ix

Upper bound

Theorem 1. For 2n ≥ 10, the graph G2n is connected and diam(G2n)≤ 5. Proof Idea.

  • large ”semiears” (≥ 12 vertices) can be rotated in at

most 3 steps

  • ”inside semicycles” can be (simultaneously) rotated in
  • ne step
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Julia Obmann Disjoint tree-compatible plane perfect matchings

7 x

Upper bound

Theorem 1. For 2n ≥ 10, the graph G2n is connected and diam(G2n)≤ 5. Proof Idea.

  • large ”semiears” (≥ 12 vertices) can be rotated in at

most 3 steps

  • ”inside semicycles” can be (simultaneously) rotated in
  • ne step
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Julia Obmann Disjoint tree-compatible plane perfect matchings

7 xi

Upper bound

Theorem 1. For 2n ≥ 10, the graph G2n is connected and diam(G2n)≤ 5. Proof Idea.

  • large ”semiears” (≥ 12 vertices) can be rotated in at

most 3 steps

  • ”inside semicycles” can be (simultaneously) rotated in
  • ne step
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Julia Obmann Disjoint tree-compatible plane perfect matchings

7 xii

Upper bound

Theorem 1. For 2n ≥ 10, the graph G2n is connected and diam(G2n)≤ 5. Proof Idea.

  • all matchings can be quickly

transformed to one of the perimeter matchings P1 or P2

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Julia Obmann Disjoint tree-compatible plane perfect matchings

7 xiii

Upper bound

Theorem 1. For 2n ≥ 10, the graph G2n is connected and diam(G2n)≤ 5. Proof Idea.

P1 P2

  • all matchings can be quickly

transformed to one of the perimeter matchings P1 or P2

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Julia Obmann Disjoint tree-compatible plane perfect matchings

7 xiv

Upper bound

  • easy bound 9:

distance to P1 or P2 is ≤ 3 for each matching Theorem 1. For 2n ≥ 10, the graph G2n is connected and diam(G2n)≤ 5. Proof Idea.

P1 P2

  • all matchings can be quickly

transformed to one of the perimeter matchings P1 or P2

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Julia Obmann Disjoint tree-compatible plane perfect matchings

7 xv

Upper bound

  • easy bound 9:

distance to P1 or P2 is ≤ 3 for each matching Theorem 1. For 2n ≥ 10, the graph G2n is connected and diam(G2n)≤ 5. Proof Idea.

P1 P2 1 1 3 2 3 3 2 S1 S2 S2 S1

P1 P2

3 4 S3

  • all matchings can be quickly

transformed to one of the perimeter matchings P1 or P2

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Julia Obmann Disjoint tree-compatible plane perfect matchings

7 xvi

Upper bound

  • easy bound 9:

distance to P1 or P2 is ≤ 3 for each matching Theorem 1. For 2n ≥ 10, the graph G2n is connected and diam(G2n)≤ 5. Proof Idea.

P1 P2 1 1 3 2 3 3 2 S1 S2 S2 S1

P1 P2

3 4 S3

  • all matchings can be quickly

transformed to one of the perimeter matchings P1 or P2

  • more sophisticated arguments

yield bound 5

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Julia Obmann Disjoint tree-compatible plane perfect matchings

8 i

Lower bound

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Julia Obmann Disjoint tree-compatible plane perfect matchings

8 ii

Lower bound

Theorem 2. For 2n ≥ 10, we have diam(G2n)≥ 4.

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Julia Obmann Disjoint tree-compatible plane perfect matchings

8 iii

Lower bound

Theorem 2. For 2n ≥ 10, we have diam(G2n)≥ 4. Idea: distance between two specific matchings is at least 4 for both n even and n odd

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Julia Obmann Disjoint tree-compatible plane perfect matchings

8 iv

Lower bound

Theorem 2. For 2n ≥ 10, we have diam(G2n)≥ 4. Idea: distance between two specific matchings is at least 4 for both n even and n odd E1 E2

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Julia Obmann Disjoint tree-compatible plane perfect matchings

8 v

Lower bound

Theorem 2. For 2n ≥ 10, we have diam(G2n)≥ 4. Idea: distance between two specific matchings is at least 4 for both n even and n odd E1 E2

g r

O1 O2

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Julia Obmann Disjoint tree-compatible plane perfect matchings

8 vi

Lower bound

Theorem 2. For 2n ≥ 10, we have diam(G2n)≥ 4. Idea: distance between two specific matchings is at least 4 for both n even and n odd (i) M tree-compatible to E1 ⇒ no green perimeter edge (analogously for E2) E1 E2

g r

O1 O2

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Julia Obmann Disjoint tree-compatible plane perfect matchings

8 vii

Lower bound

Theorem 2. For 2n ≥ 10, we have diam(G2n)≥ 4. Idea: distance between two specific matchings is at least 4 for both n even and n odd (i) M tree-compatible to E1 ⇒ no green perimeter edge (analogously for E2) E1 E2

g r

(ii) M tree-compatible to O1 ⇒ at most one green perimeter edge, which is g (analogously for O2 and r) O1 O2

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Julia Obmann Disjoint tree-compatible plane perfect matchings

8 viii

Lower bound

(iii) M and M ′ tree-compatible ⇒ at least two perimeter edges in common

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Julia Obmann Disjoint tree-compatible plane perfect matchings

8 ix

Lower bound

(i)

(iii) M and M ′ tree-compatible ⇒ at least two perimeter edges in common

(i) (iii) (iii)

E1 E2

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Julia Obmann Disjoint tree-compatible plane perfect matchings

8 x

Lower bound

g r (i)

(iii) M and M ′ tree-compatible ⇒ at least two perimeter edges in common

(i) (iii) (iii) (iii) (iii) (ii) (ii)

E1 E2 O1 O2

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Julia Obmann Disjoint tree-compatible plane perfect matchings

9 i

Summary / Open problems

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Julia Obmann Disjoint tree-compatible plane perfect matchings

9 ii

Summary / Open problems

  • The disjoint tree-compatibility graph G2n is connected

if and only if 2n ≥ 10.

  • The diameter in that case is either 4 or 5.
  • Conjecture. The diameter for all 2n ≥ 18 is 4.

(diam(G2n)= 5 for n ∈ {5, 6, 7, 8} and diam(G18)= 4)

  • Is G2n connected for general point sets (which n)?
  • Compatibility via other graph classes?

Ongoing work: disjoint path-compatibility

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Julia Obmann Disjoint tree-compatible plane perfect matchings

10

This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement No 734922.

Thanks a lot to the Organizing Committee and all involved people for adapting so well to this unusual situation and for carrying

  • ut the workshop in this way!