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

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

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

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 0 1 2 3 4 Christopher Auer | Email: christopher.auer@uni-passau.de 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 (0 , 1) 0 1 2 3 4 (0 , 3) Christopher Auer | Email: christopher.auer@uni-passau.de 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 (0 , 1) 0 1 2 3 4 (0 , 3) Christopher Auer | Email: christopher.auer@uni-passau.de 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 0 1 2 3 4 (0 , 3) Christopher Auer | Email: christopher.auer@uni-passau.de 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) 0 1 2 3 4 (0 , 3) Christopher Auer | Email: christopher.auer@uni-passau.de 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) 0 1 2 3 4 (0 , 3) Christopher Auer | Email: christopher.auer@uni-passau.de 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 0 1 2 3 4 (0 , 3) Christopher Auer | Email: christopher.auer@uni-passau.de 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 (2 , 4) 0 1 2 3 4 (0 , 3) Christopher Auer | Email: christopher.auer@uni-passau.de 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 (2 , 4) 0 1 2 3 4 (0 , 3) Christopher Auer | Email: christopher.auer@uni-passau.de 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 (2 , 4) 0 1 2 3 4 (0 , 3) Christopher Auer | Email: christopher.auer@uni-passau.de 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 (2 , 4) 0 1 2 3 4 (0 , 3) 0 1 2 4 3 Christopher Auer | Email: christopher.auer@uni-passau.de 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 ◮ Strong relationship between graph layouts and planarity (2 , 4) 0 1 2 3 4 (0 , 3) 0 1 2 4 3 Christopher Auer | Email: christopher.auer@uni-passau.de Slide 3

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

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

Introduction and Motivation Known Characterizations | Bernhart, Kainen, ’79: A graph is a. . . ◮ . . . stack graph ⇐ ⇒ it is outer-planar 0 0 1 1 4 4 2 2 3 3 Christopher Auer | Email: christopher.auer@uni-passau.de Slide 4

Introduction and Motivation Known Characterizations | Bernhart, Kainen, ’79: A graph is a. . . ◮ . . . stack graph ⇐ ⇒ it is outer-planar 0 0 1 1 4 4 2 2 3 3 0 1 2 3 4 Christopher Auer | Email: christopher.auer@uni-passau.de Slide 4

Introduction and Motivation Known Characterizations | Bernhart, Kainen, ’79: A graph is a. . . ◮ . . . stack graph ⇐ ⇒ it is outer-planar 0 0 1 1 4 4 2 2 3 3 0 1 2 3 4 Christopher Auer | Email: christopher.auer@uni-passau.de Slide 4

Introduction and Motivation Known Characterizations | Bernhart, Kainen, ’79: A graph is a. . . ◮ . . . stack graph ⇐ ⇒ it is outer-planar 0 0 1 1 4 4 2 2 3 3 0 1 2 3 4 Christopher Auer | Email: christopher.auer@uni-passau.de Slide 4

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 0 0 1 1 4 4 2 2 3 3 0 1 2 3 4 Christopher Auer | Email: christopher.auer@uni-passau.de Slide 4

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

Deque Graphs Deque Layouts | ◮ Double-ended queue Christopher Auer | Email: christopher.auer@uni-passau.de Slide 6

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

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

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

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

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

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

Deque Graphs Linear Cylindric Drawings | 4 3 2 7 6 9 5 0 1 8 Christopher Auer | Email: christopher.auer@uni-passau.de Slide 7

Deque Graphs Linear Cylindric Drawings | 4 3 2 7 6 9 5 0 1 8 0 1 2 3 4 5 6 7 8 9 Christopher Auer | Email: christopher.auer@uni-passau.de Slide 7

Deque Graphs Linear Cylindric Drawings | 4 3 2 7 6 9 5 0 1 8 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 Christopher Auer | Email: christopher.auer@uni-passau.de Slide 7

Deque Graphs Linear Cylindric Drawings | 4 3 2 7 6 9 5 0 1 8 0 1 2 3 4 5 6 7 8 9 0 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

Deque Graphs Linear Cylindric Drawings | 4 3 2 7 6 9 5 0 1 8 h h 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 t t ◮ A graph is a deque graph ⇐ ⇒ it is linear cylindric planar Christopher Auer | Email: christopher.auer@uni-passau.de Slide 7