Characterizations of Deque and Queue Graphs Christopher Auer, - - PowerPoint PPT Presentation

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Characterizations of Deque and Queue Graphs Christopher Auer, - - PowerPoint PPT Presentation

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| Characterizations of Deque and Queue Graphs Christopher Auer, Andreas Gleiner University of Passau Christopher Auer | Email: christopher.auer@uni-passau.de Slide 1 Introduction and Motivation Table of Contents | Introduction and

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SLIDE 1

|

Characterizations of Deque and Queue Graphs

Christopher Auer, Andreas Gleißner

University of Passau

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 1

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SLIDE 2

Introduction and Motivation

Table of Contents |

Introduction and Motivation Deque Graphs Proper Leveled-Planar Graphs Conclusion and Future Work

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 2

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SLIDE 3

Introduction and Motivation

Example: Stack |

◮ Graph layouts

◮ Undirected graph: G = (V , E) ◮ Linear layout π : V → {0, . . . , n − 1}: positioning of the

vertices

◮ Example: Stack layout

1 2 3 4

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 3

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SLIDE 4

Introduction and Motivation

Example: Stack |

◮ Graph layouts

◮ Undirected graph: G = (V , E) ◮ Linear layout π : V → {0, . . . , n − 1}: positioning of the

vertices

◮ Example: Stack layout

1 2 3 4 (0, 3) (0, 1)

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 3

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SLIDE 5

Introduction and Motivation

Example: Stack |

◮ Graph layouts

◮ Undirected graph: G = (V , E) ◮ Linear layout π : V → {0, . . . , n − 1}: positioning of the

vertices

◮ Example: Stack layout

1 2 3 4 (0, 3) (0, 1)

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 3

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SLIDE 6

Introduction and Motivation

Example: Stack |

◮ Graph layouts

◮ Undirected graph: G = (V , E) ◮ Linear layout π : V → {0, . . . , n − 1}: positioning of the

vertices

◮ Example: Stack layout

1 2 3 4 (0, 3)

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 3

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SLIDE 7

Introduction and Motivation

Example: Stack |

◮ Graph layouts

◮ Undirected graph: G = (V , E) ◮ Linear layout π : V → {0, . . . , n − 1}: positioning of the

vertices

◮ Example: Stack layout

1 2 3 4 (0, 3) (1, 2)

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 3

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SLIDE 8

Introduction and Motivation

Example: Stack |

◮ Graph layouts

◮ Undirected graph: G = (V , E) ◮ Linear layout π : V → {0, . . . , n − 1}: positioning of the

vertices

◮ Example: Stack layout

1 2 3 4 (0, 3) (1, 2)

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 3

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SLIDE 9

Introduction and Motivation

Example: Stack |

◮ Graph layouts

◮ Undirected graph: G = (V , E) ◮ Linear layout π : V → {0, . . . , n − 1}: positioning of the

vertices

◮ Example: Stack layout

1 2 3 4 (0, 3)

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 3

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SLIDE 10

Introduction and Motivation

Example: Stack |

◮ Graph layouts

◮ Undirected graph: G = (V , E) ◮ Linear layout π : V → {0, . . . , n − 1}: positioning of the

vertices

◮ Example: Stack layout

1 2 3 4 (0, 3) (2, 4)

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 3

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SLIDE 11

Introduction and Motivation

Example: Stack |

◮ Graph layouts

◮ Undirected graph: G = (V , E) ◮ Linear layout π : V → {0, . . . , n − 1}: positioning of the

vertices

◮ Example: Stack layout

1 2 3 4 (0, 3) (2, 4)

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 3

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SLIDE 12

Introduction and Motivation

Example: Stack |

◮ Graph layouts

◮ Undirected graph: G = (V , E) ◮ Linear layout π : V → {0, . . . , n − 1}: positioning of the

vertices

◮ Example: Stack layout

1 2 3 4 (0, 3) (2, 4)

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 3

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SLIDE 13

Introduction and Motivation

Example: Stack |

◮ Graph layouts

◮ Undirected graph: G = (V , E) ◮ Linear layout π : V → {0, . . . , n − 1}: positioning of the

vertices

◮ Example: Stack layout

1 2 3 4 (0, 3) (2, 4) 1 2 4 3

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 3

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SLIDE 14

Introduction and Motivation

Example: Stack |

◮ Graph layouts

◮ Undirected graph: G = (V , E) ◮ Linear layout π : V → {0, . . . , n − 1}: positioning of the

vertices

◮ Example: Stack layout ◮ Strong relationship between graph layouts and planarity

1 2 3 4 (0, 3) (2, 4) 1 2 4 3

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 3

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SLIDE 15

Introduction and Motivation

Known Characterizations |

Bernhart, Kainen, ’79: A graph is a. . .

◮ . . . stack graph ⇐

⇒ it is outer-planar

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 4

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SLIDE 16

Introduction and Motivation

Known Characterizations |

Bernhart, Kainen, ’79: A graph is a. . .

◮ . . . stack graph ⇐

⇒ it is outer-planar

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 4

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SLIDE 17

Introduction and Motivation

Known Characterizations |

Bernhart, Kainen, ’79: A graph is a. . .

◮ . . . stack graph ⇐

⇒ it is outer-planar 1 2 3 4 1 2 3 4

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 4

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SLIDE 18

Introduction and Motivation

Known Characterizations |

Bernhart, Kainen, ’79: A graph is a. . .

◮ . . . stack graph ⇐

⇒ it is outer-planar 1 2 3 4 1 2 3 4 1 2 3 4

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 4

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SLIDE 19

Introduction and Motivation

Known Characterizations |

Bernhart, Kainen, ’79: A graph is a. . .

◮ . . . stack graph ⇐

⇒ it is outer-planar 1 2 3 4 1 2 3 4 1 2 3 4

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 4

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SLIDE 20

Introduction and Motivation

Known Characterizations |

Bernhart, Kainen, ’79: A graph is a. . .

◮ . . . stack graph ⇐

⇒ it is outer-planar 1 2 3 4 1 2 3 4 1 2 3 4

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 4

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SLIDE 21

Introduction and Motivation

Known Characterizations |

Bernhart, Kainen, ’79: A graph is a. . .

◮ . . . stack graph ⇐

⇒ it is outer-planar

◮ . . . 2-stack graph ⇐

⇒ it is subgraph of planar graph with a Hamiltonian cycle 1 2 3 4 1 2 3 4 1 2 3 4

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 4

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SLIDE 22

Deque Graphs

Table of Contents |

Introduction and Motivation Deque Graphs Proper Leveled-Planar Graphs Conclusion and Future Work

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 5

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SLIDE 23

Deque Graphs

Deque Layouts |

◮ Double-ended queue

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 6

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SLIDE 24

Deque Graphs

Deque Layouts |

◮ Double-ended queue ◮ Two sides: Head h and Tail t

h t

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 6

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SLIDE 25

Deque Graphs

Deque Layouts |

◮ Double-ended queue ◮ Two sides: Head h and Tail t ◮ Linear I/O layout:

◮ Linear layout π : V → {0, . . . , n − 1} ◮ Input assignment α : E → {h, t} ◮ Output assignment ω : E → {h, t}

h t

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 6

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SLIDE 26

Deque Graphs

Deque Layouts |

◮ Double-ended queue ◮ Two sides: Head h and Tail t ◮ Linear I/O layout:

◮ Linear layout π : V → {0, . . . , n − 1} ◮ Input assignment α : E → {h, t} ◮ Output assignment ω : E → {h, t}

◮ α(e) = ω(e) : e is a stack edge

h t

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 6

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SLIDE 27

Deque Graphs

Deque Layouts |

◮ Double-ended queue ◮ Two sides: Head h and Tail t ◮ Linear I/O layout:

◮ Linear layout π : V → {0, . . . , n − 1} ◮ Input assignment α : E → {h, t} ◮ Output assignment ω : E → {h, t}

◮ α(e) = ω(e) : e is a stack edge ◮ α(e) = ω(e) : e is a queue edge

h t

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 6

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SLIDE 28

Deque Graphs

Deque Layouts |

◮ Double-ended queue ◮ Two sides: Head h and Tail t ◮ Linear I/O layout:

◮ Linear layout π : V → {0, . . . , n − 1} ◮ Input assignment α : E → {h, t} ◮ Output assignment ω : E → {h, t}

◮ α(e) = ω(e) : e is a stack edge ◮ α(e) = ω(e) : e is a queue edge

h t

◮ A deque. . .

◮ . . . can emulate two stacks Christopher Auer | Email: christopher.auer@uni-passau.de Slide 6

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SLIDE 29

Deque Graphs

Deque Layouts |

◮ Double-ended queue ◮ Two sides: Head h and Tail t ◮ Linear I/O layout:

◮ Linear layout π : V → {0, . . . , n − 1} ◮ Input assignment α : E → {h, t} ◮ Output assignment ω : E → {h, t}

◮ α(e) = ω(e) : e is a stack edge ◮ α(e) = ω(e) : e is a queue edge

h t

◮ A deque. . .

◮ . . . can emulate two stacks ◮ . . . allows queue items Christopher Auer | Email: christopher.auer@uni-passau.de Slide 6

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SLIDE 30

Deque Graphs

Linear Cylindric Drawings |

8 1 2 7 9 3 5 4 6

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 7

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SLIDE 31

Deque Graphs

Linear Cylindric Drawings |

8 1 2 7 9 3 5 4 6

1 2 3 4 5 6 7 8 9

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 7

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SLIDE 32

Deque Graphs

Linear Cylindric Drawings |

8 1 2 7 9 3 5 4 6

1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 7

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SLIDE 33

Deque Graphs

Linear Cylindric Drawings |

8 1 2 7 9 3 5 4 6

1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9

◮ A graph is a deque graph ⇐

⇒ it is linear cylindric planar

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 7

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SLIDE 34

Deque Graphs

Linear Cylindric Drawings |

8 1 2 7 9 3 5 4 6

1 2 3 4 5 6 7 8 9 h t 1 2 3 4 5 6 7 8 9 h t

◮ A graph is a deque graph ⇐

⇒ it is linear cylindric planar

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 7

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SLIDE 35

Deque Graphs

Linear Cylindric Drawings |

8 1 2 7 9 3 5 4 6

1 2 3 4 5 6 7 8 9 h t 1 2 3 4 5 6 7 8 9 h t

◮ A graph is a deque graph ⇐

⇒ it is linear cylindric planar

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 7

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SLIDE 36

Deque Graphs

Linear Cylindric Drawings |

8 1 2 7 9 3 5 4 6

1 2 3 4 5 6 7 8 9 h t 1 2 3 4 5 6 7 8 9 h t

◮ A graph is a deque graph ⇐

⇒ it is linear cylindric planar

◮ G is a deque graph

= ⇒ G has a planar supergraph with a Hamiltonian path

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 7

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SLIDE 37

Deque Graphs

Linear Cylindric Drawings |

8 1 2 7 9 3 5 4 6

1 2 3 4 5 6 7 8 9 h t 1 2 3 4 5 6 7 8 9 h t

◮ A graph is a deque graph ⇐

⇒ it is linear cylindric planar

◮ G is a deque graph

⇐ ⇒ G has a planar supergraph with a Hamiltonian path

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 7

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SLIDE 38

Deque Graphs

Proof Idea |

◮ Proof idea

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 8

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SLIDE 39

Deque Graphs

Proof Idea |

◮ Proof idea

◮ “Cut” along Hamiltonian path

8 1 2 7 9 3 5 4 6 8 1 2 7 9 3 5 4 6

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 8

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SLIDE 40

Deque Graphs

Proof Idea |

◮ Proof idea

◮ “Cut” along Hamiltonian path and back to start vertex

8 1 2 7 9 3 5 4 6 8 1 2 7 9 3 5 4 6

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 8

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SLIDE 41

Deque Graphs

Proof Idea |

◮ Proof idea

◮ “Cut” along Hamiltonian path and back to start vertex

8 1 2 7 9 3 5 4 6 8 1 2 7 9 3 5 4 6

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 8

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SLIDE 42

Deque Graphs

Proof Idea |

◮ Proof idea

◮ “Cut” along Hamiltonian path and back to start vertex ◮ Linear layout: Hamiltonian path

8 1 2 7 9 3 5 4 6 8 1 2 7 9 3 5 4 6

1 2 3 4 5 6 7 8 9 h t

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 8

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SLIDE 43

Deque Graphs

Proof Idea |

◮ Proof idea

◮ “Cut” along Hamiltonian path and back to start vertex ◮ Linear layout: Hamiltonian path ◮ Stack edges: within one region

8 1 2 7 9 3 5 4 6 8 1 2 7 9 3 5 4 6

1 2 3 4 5 6 7 8 9 h t

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 8

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SLIDE 44

Deque Graphs

Proof Idea |

◮ Proof idea

◮ “Cut” along Hamiltonian path and back to start vertex ◮ Linear layout: Hamiltonian path ◮ Stack edges: within one region ◮ Queue edges: different regions

8 1 2 7 9 3 5 4 6 8 1 2 7 9 3 5 4 6

1 2 3 4 5 6 7 8 9 h t

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 8

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SLIDE 45

Deque Graphs

Queue Graphs | Christopher Auer | Email: christopher.auer@uni-passau.de Slide 9

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SLIDE 46

Deque Graphs

Queue Graphs |

1 2 3 4 1 2 3 4

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 9

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SLIDE 47

Deque Graphs

Queue Graphs |

1 2 3 4 1 2 3 4

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 9

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SLIDE 48

Deque Graphs

Queue Graphs |

1 2 3 4 1 2 3 4

◮ All edges are queue edges =

⇒ queue graph

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 9

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SLIDE 49

Deque Graphs

Queue Graphs |

1 2 3 4 1 2 3 4

◮ All edges are queue edges =

⇒ queue graph

◮ Consequence: The dual of an embedded queue graph has a

Eularian path

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 9

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SLIDE 50

Proper Leveled-Planar Graphs

Table of Contents |

Introduction and Motivation Deque Graphs Proper Leveled-Planar Graphs Conclusion and Future Work

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 10

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SLIDE 51

Proper Leveled-Planar Graphs

Arched Leveled-Planar Graphs |

◮ G is a queue graph ⇐

⇒ G is arched leveled-planar (Heath and Rosenberg)

1 2 3 4 5 6

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 11

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SLIDE 52

Proper Leveled-Planar Graphs

Arched Leveled-Planar Graphs |

◮ G is a queue graph ⇐

⇒ G is arched leveled-planar (Heath and Rosenberg)

◮ “almost” proper leveled-planar 1 2 3 4 5 6

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 11

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SLIDE 53

Proper Leveled-Planar Graphs

Arched Leveled-Planar Graphs |

◮ G is a queue graph ⇐

⇒ G is arched leveled-planar (Heath and Rosenberg)

◮ “almost” proper leveled-planar 1 2 3 4 5 6

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 11

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SLIDE 54

Proper Leveled-Planar Graphs

Arched Leveled-Planar Graphs |

◮ G is a queue graph ⇐

⇒ G is arched leveled-planar (Heath and Rosenberg)

◮ “almost” proper leveled-planar 1 2 3 4 5 6

Proposition: G is proper leveled-planar = ⇒ G is a bipartite queue graph

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 11

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SLIDE 55

Proper Leveled-Planar Graphs

Arched Leveled-Planar Graphs |

◮ G is a queue graph ⇐

⇒ G is arched leveled-planar (Heath and Rosenberg)

◮ “almost” proper leveled-planar ◮ arched leveled-planar bipartite 1 2 3 4 5 6 1 2

Proposition: G is proper leveled-planar = ⇒ G is a bipartite queue graph

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 11

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SLIDE 56

Proper Leveled-Planar Graphs

Arched Leveled-Planar Graphs |

◮ G is a queue graph ⇐

⇒ G is arched leveled-planar (Heath and Rosenberg)

◮ “almost” proper leveled-planar ◮ arched leveled-planar bipartite 1 2 3 4 5 6 1 2

Theorem: G is proper leveled-planar ⇐ ⇒ G is a bipartite queue graph

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 11

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SLIDE 57

Proper Leveled-Planar Graphs

Proof Idea |

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 12

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SLIDE 58

Proper Leveled-Planar Graphs

Proof Idea |

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 12

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SLIDE 59

Proper Leveled-Planar Graphs

Proof Idea |

A D E C B 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 12

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SLIDE 60

Proper Leveled-Planar Graphs

Proof Idea |

A D E C B 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

A C B D E

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 12

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SLIDE 61

Proper Leveled-Planar Graphs

Proof Idea |

A D E C B 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

A C B D E

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 12

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SLIDE 62

Proper Leveled-Planar Graphs

Proof Idea |

A D E C B 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

A C B D E

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 12

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SLIDE 63

Proper Leveled-Planar Graphs

Proof Idea |

A D E C B 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 1 6

A C B D E A

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 12

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SLIDE 64

Proper Leveled-Planar Graphs

Proof Idea |

A D E C B 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 1 6 3 8 10 12 13 15

A C B D E A B

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 12

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SLIDE 65

Proper Leveled-Planar Graphs

Proof Idea |

A D E C B 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 1 6 3 8 10 12 13 15 2 5 7 11

A C B D E A B C

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 12

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SLIDE 66

Proper Leveled-Planar Graphs

Proof Idea |

A D E C B 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 1 6 3 8 10 12 13 15 2 5 7 11 14 16 17

A C B D E A B C E

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 12

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SLIDE 67

Proper Leveled-Planar Graphs

Proof Idea |

A D E C B 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 1 6 3 8 10 12 13 15 2 5 7 11 14 16 17 4 9

A C B D E A B C E D

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 12

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SLIDE 68

Conclusion and Future Work

Table of Contents |

Introduction and Motivation Deque Graphs Proper Leveled-Planar Graphs Conclusion and Future Work

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 13

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SLIDE 69

Conclusion and Future Work

Conclusion and Future Work |

◮ Deque layouts

◮ Planar and Hamiltonian cycle ⇐

⇒ 2-Stacks

◮ Planar and Hamiltonian path ⇐

⇒ Deque

1 2 3 4 5 6 7 8 9

1 2 3 4 5 6

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 14

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SLIDE 70

Conclusion and Future Work

Conclusion and Future Work |

◮ Deque layouts

◮ Planar and Hamiltonian cycle ⇐

⇒ 2-Stacks

◮ Planar and Hamiltonian path ⇐

⇒ Deque

◮ Future: Planar ⇐

⇒ Extended Deque

1 2 3 4 5 6 7 8 9

1 2 3 4 5 6

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 14

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SLIDE 71

Conclusion and Future Work

Conclusion and Future Work |

◮ Deque layouts

◮ Planar and Hamiltonian cycle ⇐

⇒ 2-Stacks

◮ Planar and Hamiltonian path ⇐

⇒ Deque

◮ Future: Planar ⇐

⇒ Extended Deque

◮ Proper leveled-planar ⇐

⇒ Queue and Bipartite

1 2 3 4 5 6 7 8 9

1 2 3 4 5 6

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 14

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SLIDE 72

Conclusion and Future Work

Conclusion and Future Work |

◮ Deque layouts

◮ Planar and Hamiltonian cycle ⇐

⇒ 2-Stacks

◮ Planar and Hamiltonian path ⇐

⇒ Deque

◮ Future: Planar ⇐

⇒ Extended Deque

◮ Proper leveled-planar ⇐

⇒ Queue and Bipartite

◮ Dual of embedded queue graph contains

Eulerian path

1 2 3 4 5 6 7 8 9

1 2 3 4 5 6

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 14

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SLIDE 73

Conclusion and Future Work

Conclusion and Future Work |

◮ Deque layouts

◮ Planar and Hamiltonian cycle ⇐

⇒ 2-Stacks

◮ Planar and Hamiltonian path ⇐

⇒ Deque

◮ Future: Planar ⇐

⇒ Extended Deque

◮ Proper leveled-planar ⇐

⇒ Queue and Bipartite

◮ Dual of embedded queue graph contains

Eulerian path

◮ Respective decision problems: all NP-complete

1 2 3 4 5 6 7 8 9

1 2 3 4 5 6

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 14

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SLIDE 74

Conclusion and Future Work

Conclusion and Future Work |

◮ Deque layouts

◮ Planar and Hamiltonian cycle ⇐

⇒ 2-Stacks

◮ Planar and Hamiltonian path ⇐

⇒ Deque

◮ Future: Planar ⇐

⇒ Extended Deque

◮ Proper leveled-planar ⇐

⇒ Queue and Bipartite

◮ Dual of embedded queue graph contains

Eulerian path

◮ Respective decision problems: all NP-complete ◮ Heath and Rosenberg conjectured: “Every

planar graph is a queue and stack graph”

1 2 3 4 5 6 7 8 9

1 2 3 4 5 6

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 14

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SLIDE 75

Conclusion and Future Work

Conclusion and Future Work |

◮ Deque layouts

◮ Planar and Hamiltonian cycle ⇐

⇒ 2-Stacks

◮ Planar and Hamiltonian path ⇐

⇒ Deque

◮ Future: Planar ⇐

⇒ Extended Deque

◮ Proper leveled-planar ⇐

⇒ Queue and Bipartite

◮ Dual of embedded queue graph contains

Eulerian path

◮ Respective decision problems: all NP-complete ◮ Heath and Rosenberg conjectured: “Every

planar graph is a queue and stack graph”

◮ Our conjecture: this is not true

1 2 3 4 5 6 7 8 9

1 2 3 4 5 6

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 14

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SLIDE 76

Conclusion and Future Work

Conclusion and Future Work |

◮ Deque layouts

◮ Planar and Hamiltonian cycle ⇐

⇒ 2-Stacks

◮ Planar and Hamiltonian path ⇐

⇒ Deque

◮ Future: Planar ⇐

⇒ Extended Deque

◮ Proper leveled-planar ⇐

⇒ Queue and Bipartite

◮ Dual of embedded queue graph contains

Eulerian path

◮ Respective decision problems: all NP-complete ◮ Heath and Rosenberg conjectured: “Every

planar graph is a queue and stack graph”

◮ Our conjecture: this is not true

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1 2 3 4 5 6

T h a n k Y

  • u

!

Christopher Auer | Email: christopher.auer@uni-passau.de Slide 14