[PPT] - On Strict (Outer-)Confluent Graphs Henry F orster, Robert Ganian, PowerPoint Presentation

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Graph Drawing 2019 · September 18, 2019

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg

On Strict (Outer-)Confluent Graphs

SLIDE 2

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

2/21

Confluence

Technique to bundle edges

SLIDE 3

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

2/21

Confluence

Technique to bundle edges

SLIDE 4

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

2/21

Confluence

Technique to bundle edges

SLIDE 5

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

2/21

Confluence

Technique to bundle edges

SLIDE 6

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

2/21

Confluence

Technique to bundle edges

SLIDE 7

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

2/21

Confluence

Technique to bundle edges

SLIDE 8

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

2/21

Confluence

Technique to bundle edges

SLIDE 9

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

2/21

Confluence

Technique to bundle edges

SLIDE 10

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

3/21

Definitions

Plane drawing, i.e. no “real” intersections No wrong adjacencies No double paths ⇒ strict confluency All vertices on a circle ⇒ outer confluency

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Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

3/21

Definitions

Junction Plane drawing, i.e. no “real” intersections No wrong adjacencies No double paths ⇒ strict confluency All vertices on a circle ⇒ outer confluency Node

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Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

3/21

Definitions

Junction Arc Plane drawing, i.e. no “real” intersections No wrong adjacencies No double paths ⇒ strict confluency All vertices on a circle ⇒ outer confluency Node

SLIDE 13

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

3/21

Definitions

Junction Arc Path Plane drawing, i.e. no “real” intersections No wrong adjacencies No double paths ⇒ strict confluency All vertices on a circle ⇒ outer confluency Node

SLIDE 14

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

3/21

Definitions

Junction Arc Path Plane drawing, i.e. no “real” intersections No wrong adjacencies No double paths ⇒ strict confluency All vertices on a circle ⇒ outer confluency Node Every plane drawing is (strict) confluent ⇒ Questions mainly interesting for non-planar graphs

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Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

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Known Results

Only known for a few classes of graphs E.g. Interval graphs, bipartite permutation graphs [Dickerson et al 2005, Hui et al. 2007] Negative case also only known for a few classes of graphs E.g. Petersen graph, Chordal graphs [Dickerson et al. 2005] Does a graph G admit a confluent drawing? Recognizing strict-confluent graphs is NP-hard [Eppstein et al. 2016]

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Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

4/21

Known Results

One main open question Which graphs have a strict outer-confluent drawing? Only known for a few classes of graphs E.g. Interval graphs, bipartite permutation graphs [Dickerson et al 2005, Hui et al. 2007] Negative case also only known for a few classes of graphs E.g. Petersen graph, Chordal graphs [Dickerson et al. 2005] Does a graph G admit a confluent drawing? Recognizing strict-confluent graphs is NP-hard [Eppstein et al. 2016]

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Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

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SC Drawings of Unit Interval Graphs

v1 v4 w1 w5 x1 x2 Graphs that can be represented as intersection of unit intervals

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Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

5/21

SC Drawings of Unit Interval Graphs

C1 C3 v1 v4 w1 C2 w5 x1 x2 Graphs that can be represented as intersection of unit intervals Decompose into cliques

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Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

5/21

SC Drawings of Unit Interval Graphs

C1 C3 v1 v4 w1 C2 w5 x1 x2 v1 v2 v3 v4 x1 x2 d2 d2 d3 d3 d4 w1 w2 w3 w4 w5 Graphs that can be represented as intersection of unit intervals Decompose into cliques

SLIDE 20

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

5/21

SC Drawings of Unit Interval Graphs

C1 C3 v1 v4 w1 C2 w5 x1 x2 v1 v2 v3 v4 x1 x2 d2 d2 b2 d3 d3 d4 b3 b4 w1 w2 w3 w4 w5 b4 b3 Graphs that can be represented as intersection of unit intervals Decompose into cliques

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Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

5/21

SC Drawings of Unit Interval Graphs

C1 C3 v1 v4 w1 C2 w5 x1 x2 v1 v2 v3 v4 x1 x2 d2 d2 b2 H d3 d3 d4 b3 b4 w1 w2 w3 w4 w5 b4 b3 br Graphs that can be represented as intersection of unit intervals Decompose into cliques Connect the cliques with confluent paths

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Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

6/21

Our Results

strict-outerconfluent strict-confluent

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Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

6/21

Our Results

unit interval strict-outerconfluent strict-confluent

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Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

6/21

Our Results

unit interval strict-outerconfluent strict-confluent strict bipartite outerconfluent bipartite permutation ∩ domino-free

SLIDE 25

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

6/21

Our Results

unit interval strict-outerconfluent strict-confluent strict bipartite outerconfluent bipartite permutation ∩ domino-free

SLIDE 26

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

6/21

Our Results

unit interval strict-outerconfluent strict-confluent strict bipartite outerconfluent bipartite permutation ∩ domino-free

SLIDE 27

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

6/21

Our Results

unit interval strict-outerconfluent strict-confluent strict bipartite outerconfluent bipartite permutation ∩ domino-free bipartite permutation

SLIDE 28

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

6/21

Our Results

unit interval strict-outerconfluent strict-confluent strict bipartite outerconfluent bipartite permutation ∩ domino-free bipartite permutation bipartite permutation

SLIDE 29

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

6/21

Our Results

unit interval strict-outerconfluent strict-confluent strict bipartite outerconfluent bipartite permutation ∩ domino-free bipartite permutation bipartite permutation

SLIDE 30

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

6/21

Our Results

unit interval strict-outerconfluent strict-confluent strict bipartite outerconfluent bipartite permutation ∩ domino-free bipartite permutation comparability circle trapezoid circular arc bipartite permutation

SLIDE 31

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

6/21

Our Results

unit interval strict-outerconfluent strict-confluent strict bipartite outerconfluent bipartite permutation ∩ domino-free bipartite permutation comparability circle trapezoid circular arc circle-trapezoid bipartite permutation

SLIDE 32

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

6/21

Our Results

unit interval strict-outerconfluent strict-confluent strict bipartite outerconfluent bipartite permutation ∩ domino-free bipartite permutation comparability circle trapezoid circular arc circle-trapezoid series-parallel pseudo-split chordal polygon-circle bipartite permutation

SLIDE 33

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

6/21

Our Results

unit interval strict-outerconfluent strict-confluent strict bipartite outerconfluent bipartite permutation ∩ domino-free bipartite permutation comparability circle trapezoid circular arc circle-trapezoid series-parallel pseudo-split chordal polygon-circle interval-filament co-comparability bipartite permutation

SLIDE 34

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

6/21

Our Results

unit interval strict-outerconfluent strict-confluent strict bipartite outerconfluent bipartite permutation ∩ domino-free bipartite permutation comparability circle trapezoid circular arc circle-trapezoid series-parallel pseudo-split chordal polygon-circle interval-filament co-comparability subtree-filament bipartite permutation

SLIDE 35

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

6/21

Our Results

unit interval strict-outerconfluent strict-confluent strict bipartite outerconfluent bipartite permutation ∩ domino-free string

uter-string

bipartite permutation comparability circle trapezoid circular arc circle-trapezoid series-parallel pseudo-split chordal polygon-circle interval-filament co-comparability subtree-filament bipartite permutation

SLIDE 36

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

6/21

Our Results

unit interval strict-outerconfluent strict-confluent strict bipartite outerconfluent bipartite permutation ∩ domino-free string

uter-string

bipartite permutation comparability circle trapezoid circular arc circle-trapezoid series-parallel pseudo-split chordal polygon-circle interval-filament co-comparability subtree-filament bipartite permutation

SLIDE 37

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

6/21

Our Results

unit interval strict-outerconfluent strict-confluent strict bipartite outerconfluent bipartite permutation ∩ domino-free string

uter-string

bipartite permutation comparability circle trapezoid circular arc circle-trapezoid series-parallel pseudo-split chordal polygon-circle interval-filament co-comparability subtree-filament bipartite permutation

SLIDE 38

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

6/21

Our Results

unit interval strict-outerconfluent strict-confluent strict bipartite outerconfluent bipartite permutation ∩ domino-free string

uter-string

bipartite permutation comparability circle trapezoid circular arc circle-trapezoid series-parallel pseudo-split chordal polygon-circle interval-filament co-comparability subtree-filament ∆-confluent bipartite permutation

SLIDE 39

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

6/21

Our Results

unit interval strict-outerconfluent strict-confluent strict bipartite outerconfluent bipartite permutation ∩ domino-free string

uter-string

bipartite permutation comparability circle trapezoid circular arc circle-trapezoid series-parallel pseudo-split chordal polygon-circle interval-filament co-comparability subtree-filament distance-hereditary ∆-confluent bipartite permutation

SLIDE 40

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

6/21

Our Results

unit interval strict-outerconfluent strict-confluent strict bipartite outerconfluent bipartite permutation ∩ domino-free string

uter-string

bipartite permutation comparability circle trapezoid circular arc circle-trapezoid series-parallel pseudo-split chordal polygon-circle interval-filament co-comparability subtree-filament distance-hereditary ∆-confluent tree-like ∆-strict-outerconfluent bipartite permutation

SLIDE 41

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

6/21

Our Results

unit interval strict-outerconfluent strict-confluent strict bipartite outerconfluent bipartite permutation ∩ domino-free string

uter-string

bipartite permutation comparability circle trapezoid circular arc circle-trapezoid series-parallel pseudo-split chordal polygon-circle interval-filament co-comparability subtree-filament distance-hereditary ∆-confluent tree-like ∆-strict-outerconfluent bipartite permutation

SLIDE 42

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

6/21

Our Results

unit interval strict-outerconfluent strict-confluent strict bipartite outerconfluent bipartite permutation ∩ domino-free string

uter-string

bipartite permutation comparability circle trapezoid circular arc circle-trapezoid series-parallel pseudo-split chordal polygon-circle interval-filament co-comparability subtree-filament distance-hereditary ∆-confluent tree-like ∆-strict-outerconfluent bipartite permutation

SLIDE 43

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

6/21

Our Results

unit interval strict-outerconfluent strict-confluent strict bipartite outerconfluent bipartite permutation ∩ domino-free string

uter-string

bipartite permutation comparability circle trapezoid circular arc circle-trapezoid series-parallel pseudo-split chordal polygon-circle interval-filament co-comparability subtree-filament distance-hereditary ∆-confluent tree-like ∆-strict-outerconfluent

bounded cliquewidth

bipartite permutation

SLIDE 44

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

6/21

Our Results

unit interval strict-outerconfluent strict-confluent strict bipartite outerconfluent bipartite permutation ∩ domino-free string

uter-string

bipartite permutation comparability circle trapezoid circular arc circle-trapezoid series-parallel pseudo-split chordal polygon-circle interval-filament co-comparability subtree-filament distance-hereditary ∆-confluent tree-like ∆-strict-outerconfluent

bounded cliquewidth

bipartite permutation

SLIDE 45

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

6/21

Our Results

unit interval strict-outerconfluent strict-confluent strict bipartite outerconfluent bipartite permutation ∩ domino-free string

uter-string

bipartite permutation comparability circle trapezoid circular arc circle-trapezoid series-parallel pseudo-split chordal polygon-circle interval-filament co-comparability subtree-filament distance-hereditary ∆-confluent tree-like ∆-strict-outerconfluent

bounded cliquewidth

bipartite permutation

?

SLIDE 46

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

6/21

Our Results

unit interval strict-outerconfluent strict-confluent strict bipartite outerconfluent bipartite permutation ∩ domino-free string

uter-string

bipartite permutation comparability circle trapezoid circular arc circle-trapezoid series-parallel pseudo-split chordal polygon-circle interval-filament co-comparability subtree-filament distance-hereditary ∆-confluent tree-like ∆-strict-outerconfluent

bounded cliquewidth

bipartite permutation

? Cop number: c [Gavenˇ

ciak et al. 18]

SLIDE 47

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

6/21

Our Results

unit interval strict-outerconfluent strict-confluent strict bipartite outerconfluent bipartite permutation ∩ domino-free string

uter-string

bipartite permutation comparability circle trapezoid circular arc circle-trapezoid series-parallel pseudo-split chordal polygon-circle interval-filament co-comparability subtree-filament distance-hereditary ∆-confluent tree-like ∆-strict-outerconfluent

bounded cliquewidth

bipartite permutation

? Cop number: c [Gavenˇ

ciak et al. 18]

3 ≤ c ≤ 15

SLIDE 48

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

6/21

Our Results

unit interval strict-outerconfluent strict-confluent strict bipartite outerconfluent bipartite permutation ∩ domino-free string

uter-string

bipartite permutation comparability circle trapezoid circular arc circle-trapezoid series-parallel pseudo-split chordal polygon-circle interval-filament co-comparability subtree-filament distance-hereditary ∆-confluent tree-like ∆-strict-outerconfluent

bounded cliquewidth

bipartite permutation

? Cop number: c [Gavenˇ

ciak et al. 18]

3 ≤ c ≤ 15 3 ≤ c ≤ 4

SLIDE 49

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

6/21

Our Results

unit interval strict-outerconfluent strict-confluent strict bipartite outerconfluent bipartite permutation ∩ domino-free string

uter-string

bipartite permutation comparability circle trapezoid circular arc circle-trapezoid series-parallel pseudo-split chordal polygon-circle interval-filament co-comparability subtree-filament distance-hereditary ∆-confluent tree-like ∆-strict-outerconfluent

bounded cliquewidth

bipartite permutation

? Cop number: c [Gavenˇ

ciak et al. 18]

3 ≤ c ≤ 15 3 ≤ c ≤ 4 c = 2

SLIDE 50

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

6/21

Our Results

unit interval strict-outerconfluent strict-confluent strict bipartite outerconfluent bipartite permutation ∩ domino-free string

uter-string

bipartite permutation comparability circle trapezoid circular arc circle-trapezoid series-parallel pseudo-split chordal polygon-circle interval-filament co-comparability subtree-filament distance-hereditary ∆-confluent tree-like ∆-strict-outerconfluent

bounded cliquewidth

bipartite permutation

? Cop number: c [Gavenˇ

ciak et al. 18]

3 ≤ c ≤ 15 3 ≤ c ≤ 4 c = 2 c = 1 c = 1

SLIDE 51

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

6/21

Our Results

unit interval strict-outerconfluent strict-confluent strict bipartite outerconfluent bipartite permutation ∩ domino-free string

uter-string

bipartite permutation comparability circle trapezoid circular arc circle-trapezoid series-parallel pseudo-split chordal polygon-circle interval-filament co-comparability subtree-filament distance-hereditary ∆-confluent tree-like ∆-strict-outerconfluent

bounded cliquewidth

bipartite permutation

? Cop number: c [Gavenˇ

ciak et al. 18]

3 ≤ c ≤ 15 3 ≤ c ≤ 4 c = 2 c = 2 c = 1 c = 1

SLIDE 52

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

6/21

Our Results

unit interval strict-outerconfluent strict-confluent strict bipartite outerconfluent bipartite permutation ∩ domino-free string

uter-string

bipartite permutation comparability circle trapezoid circular arc circle-trapezoid series-parallel pseudo-split chordal polygon-circle interval-filament co-comparability subtree-filament distance-hereditary ∆-confluent tree-like ∆-strict-outerconfluent

bounded cliquewidth

bipartite permutation

? Cop number: c [Gavenˇ

ciak et al. 18]

3 ≤ c ≤ 15 3 ≤ c ≤ 4 c = 2 c = 2

?

c = 1 c = 1

SLIDE 53

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

6/21

Our Results

unit interval strict-outerconfluent strict-confluent strict bipartite outerconfluent bipartite permutation ∩ domino-free string

uter-string

bipartite permutation comparability circle trapezoid circular arc circle-trapezoid series-parallel pseudo-split chordal polygon-circle interval-filament co-comparability subtree-filament distance-hereditary ∆-confluent tree-like ∆-strict-outerconfluent

bounded cliquewidth

bipartite permutation

? Cop number: c [Gavenˇ

ciak et al. 18]

3 ≤ c ≤ 15 3 ≤ c ≤ 4 c = 2 c = 2

?

c = 1 c = 1

SLIDE 54

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

7/21

Cops and Robbers

Simple two player game on a graph Cop player has k cop tokens Robber player has 1 robber token In a turn player can move their token to adjacent vertices Robber is caught if it coincides with a cop Cop-number: What is the smallest k such that cop player wins?

SLIDE 55

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

7/21

Cops and Robbers

Simple two player game on a graph Cop player has k cop tokens Robber player has 1 robber token In a turn player can move their token to adjacent vertices Robber is caught if it coincides with a cop Cop-number: What is the smallest k such that cop player wins? Example: A path

SLIDE 56

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

7/21

Cops and Robbers

Simple two player game on a graph Cop player has k cop tokens Robber player has 1 robber token In a turn player can move their token to adjacent vertices Robber is caught if it coincides with a cop Cop-number: What is the smallest k such that cop player wins? Example: A path

SLIDE 57

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

7/21

Cops and Robbers

Simple two player game on a graph Cop player has k cop tokens Robber player has 1 robber token In a turn player can move their token to adjacent vertices Robber is caught if it coincides with a cop Cop-number: What is the smallest k such that cop player wins? Example: A path

SLIDE 58

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

7/21

Cops and Robbers

Simple two player game on a graph Cop player has k cop tokens Robber player has 1 robber token In a turn player can move their token to adjacent vertices Robber is caught if it coincides with a cop Cop-number: What is the smallest k such that cop player wins? Example: A path

SLIDE 59

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

7/21

Cops and Robbers

Simple two player game on a graph Cop player has k cop tokens Robber player has 1 robber token In a turn player can move their token to adjacent vertices Robber is caught if it coincides with a cop Cop-number: What is the smallest k such that cop player wins? Example: A path

SLIDE 60

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

7/21

Cops and Robbers

Simple two player game on a graph Cop player has k cop tokens Robber player has 1 robber token In a turn player can move their token to adjacent vertices Robber is caught if it coincides with a cop Cop-number: What is the smallest k such that cop player wins? Example: A path

SLIDE 61

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

7/21

Cops and Robbers

Simple two player game on a graph Cop player has k cop tokens Robber player has 1 robber token In a turn player can move their token to adjacent vertices Robber is caught if it coincides with a cop Cop-number: What is the smallest k such that cop player wins? Example: A path

SLIDE 62

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

7/21

Cops and Robbers

Simple two player game on a graph Cop player has k cop tokens Robber player has 1 robber token In a turn player can move their token to adjacent vertices Robber is caught if it coincides with a cop Cop-number: What is the smallest k such that cop player wins? Example: A path

SLIDE 63

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

7/21

Cops and Robbers

Simple two player game on a graph Cop player has k cop tokens Robber player has 1 robber token In a turn player can move their token to adjacent vertices Robber is caught if it coincides with a cop Cop-number: What is the smallest k such that cop player wins? Example: A path

SLIDE 64

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

7/21

Cops and Robbers

Simple two player game on a graph Cop player has k cop tokens Robber player has 1 robber token In a turn player can move their token to adjacent vertices Robber is caught if it coincides with a cop Cop-number: What is the smallest k such that cop player wins? Example: A path

SLIDE 65

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

7/21

Cops and Robbers

Simple two player game on a graph Cop player has k cop tokens Robber player has 1 robber token In a turn player can move their token to adjacent vertices Robber is caught if it coincides with a cop Cop-number: What is the smallest k such that cop player wins? Example: A path

SLIDE 66

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

7/21

Cops and Robbers

Simple two player game on a graph Cop player has k cop tokens Robber player has 1 robber token In a turn player can move their token to adjacent vertices Robber is caught if it coincides with a cop Cop-number: What is the smallest k such that cop player wins? Example: A circle

SLIDE 67

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

7/21

Cops and Robbers

Simple two player game on a graph Cop player has k cop tokens Robber player has 1 robber token In a turn player can move their token to adjacent vertices Robber is caught if it coincides with a cop Cop-number: What is the smallest k such that cop player wins? Example: A circle

SLIDE 68

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

7/21

Cops and Robbers

Simple two player game on a graph Cop player has k cop tokens Robber player has 1 robber token In a turn player can move their token to adjacent vertices Robber is caught if it coincides with a cop Cop-number: What is the smallest k such that cop player wins? Example: A circle

SLIDE 69

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

7/21

Cops and Robbers

Simple two player game on a graph Cop player has k cop tokens Robber player has 1 robber token In a turn player can move their token to adjacent vertices Robber is caught if it coincides with a cop Cop-number: What is the smallest k such that cop player wins? Example: A circle

SLIDE 70

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

7/21

Cops and Robbers

Simple two player game on a graph Cop player has k cop tokens Robber player has 1 robber token In a turn player can move their token to adjacent vertices Robber is caught if it coincides with a cop Cop-number: What is the smallest k such that cop player wins? Example: A circle

SLIDE 71

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

7/21

Cops and Robbers

Simple two player game on a graph Cop player has k cop tokens Robber player has 1 robber token In a turn player can move their token to adjacent vertices Robber is caught if it coincides with a cop Cop-number: What is the smallest k such that cop player wins? Example: A circle

SLIDE 72

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

7/21

Cops and Robbers

Simple two player game on a graph Cop player has k cop tokens Robber player has 1 robber token In a turn player can move their token to adjacent vertices Robber is caught if it coincides with a cop Cop-number: What is the smallest k such that cop player wins? Example: A circle

SLIDE 73

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

7/21

Cops and Robbers

Simple two player game on a graph Cop player has k cop tokens Robber player has 1 robber token In a turn player can move their token to adjacent vertices Robber is caught if it coincides with a cop Cop-number: What is the smallest k such that cop player wins? Example: A circle

SLIDE 74

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

7/21

Cops and Robbers

Simple two player game on a graph Cop player has k cop tokens Robber player has 1 robber token In a turn player can move their token to adjacent vertices Robber is caught if it coincides with a cop Cop-number: What is the smallest k such that cop player wins? Example: A circle

SLIDE 75

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

7/21

Cops and Robbers

Simple two player game on a graph Cop player has k cop tokens Robber player has 1 robber token In a turn player can move their token to adjacent vertices Robber is caught if it coincides with a cop Cop-number: What is the smallest k such that cop player wins? Example: A circle

SLIDE 76

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

7/21

Cops and Robbers

Simple two player game on a graph Cop player has k cop tokens Robber player has 1 robber token In a turn player can move their token to adjacent vertices Robber is caught if it coincides with a cop Cop-number: What is the smallest k such that cop player wins? Example: A circle

SLIDE 77

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

7/21

Cops and Robbers

Simple two player game on a graph Cop player has k cop tokens Robber player has 1 robber token In a turn player can move their token to adjacent vertices Robber is caught if it coincides with a cop Cop-number: What is the smallest k such that cop player wins? Example: A circle

SLIDE 78

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

7/21

Cops and Robbers

Simple two player game on a graph Cop player has k cop tokens Robber player has 1 robber token In a turn player can move their token to adjacent vertices Robber is caught if it coincides with a cop Cop-number: What is the smallest k such that cop player wins? Example: A circle

SLIDE 79

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

7/21

Cops and Robbers

Simple two player game on a graph Cop player has k cop tokens Robber player has 1 robber token In a turn player can move their token to adjacent vertices Robber is caught if it coincides with a cop Cop-number: What is the smallest k such that cop player wins? Example: A circle

SLIDE 80

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

7/21

Cops and Robbers

Simple two player game on a graph Cop player has k cop tokens Robber player has 1 robber token In a turn player can move their token to adjacent vertices Robber is caught if it coincides with a cop Cop-number: What is the smallest k such that cop player wins? Example: A circle

SLIDE 81

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

7/21

Cops and Robbers

Simple two player game on a graph Cop player has k cop tokens Robber player has 1 robber token In a turn player can move their token to adjacent vertices Robber is caught if it coincides with a cop Cop-number: What is the smallest k such that cop player wins? Example: A circle

SLIDE 82

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

7/21

Cops and Robbers

Simple two player game on a graph Cop player has k cop tokens Robber player has 1 robber token In a turn player can move their token to adjacent vertices Robber is caught if it coincides with a cop Cop-number: What is the smallest k such that cop player wins? Example: A circle

SLIDE 83

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

7/21

Cops and Robbers

Simple two player game on a graph Cop player has k cop tokens Robber player has 1 robber token In a turn player can move their token to adjacent vertices Robber is caught if it coincides with a cop Cop-number: What is the smallest k such that cop player wins? Example: A circle

SLIDE 84

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

7/21

Cops and Robbers

Simple two player game on a graph Cop player has k cop tokens Robber player has 1 robber token In a turn player can move their token to adjacent vertices Robber is caught if it coincides with a cop Cop-number: What is the smallest k such that cop player wins? Example: A circle

SLIDE 85

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

7/21

Cops and Robbers

Simple two player game on a graph Cop player has k cop tokens Robber player has 1 robber token In a turn player can move their token to adjacent vertices Robber is caught if it coincides with a cop Cop-number: What is the smallest k such that cop player wins? Example: A circle

SLIDE 86

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

7/21

Cops and Robbers

Simple two player game on a graph Cop player has k cop tokens Robber player has 1 robber token In a turn player can move their token to adjacent vertices Robber is caught if it coincides with a cop Cop-number: What is the smallest k such that cop player wins? Example: A circle

SLIDE 87

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

7/21

Cops and Robbers

Simple two player game on a graph Cop player has k cop tokens Robber player has 1 robber token In a turn player can move their token to adjacent vertices Robber is caught if it coincides with a cop Cop-number: What is the smallest k such that cop player wins? Example: A circle

SLIDE 88

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

8/21

Cops and Robbers meeting SOC Graphs

Result: Strict outer-confluent graphs have cop-number 2

SLIDE 89

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

8/21

Cops and Robbers meeting SOC Graphs

Result: Strict outer-confluent graphs have cop-number 2 v Using one cop, robber is “locked” between u and v r u

SLIDE 90

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

8/21

Cops and Robbers meeting SOC Graphs

Result: Strict outer-confluent graphs have cop-number 2 v Using one cop, robber is “locked” between u and v r u

SLIDE 91

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

8/21

Cops and Robbers meeting SOC Graphs

Result: Strict outer-confluent graphs have cop-number 2 v Using one cop, robber is “locked” between u and v r u

SLIDE 92

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

8/21

Cops and Robbers meeting SOC Graphs

Result: Strict outer-confluent graphs have cop-number 2 v Using one cop, robber is “locked” between u and v r u

SLIDE 93

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

8/21

Cops and Robbers meeting SOC Graphs

Result: Strict outer-confluent graphs have cop-number 2 v Using one cop, robber is “locked” between u and v r u

SLIDE 94

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

8/21

Cops and Robbers meeting SOC Graphs

Result: Strict outer-confluent graphs have cop-number 2 v Using one cop, robber is “locked” between u and v r u

SLIDE 95

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

8/21

Cops and Robbers meeting SOC Graphs

Result: Strict outer-confluent graphs have cop-number 2 v Using one cop, robber is “locked” between u and v r u

SLIDE 96

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

8/21

Cops and Robbers meeting SOC Graphs

Result: Strict outer-confluent graphs have cop-number 2 v Using one cop, robber is “locked” between u and v r u

SLIDE 97

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

8/21

Cops and Robbers meeting SOC Graphs

Result: Strict outer-confluent graphs have cop-number 2 v Using one cop, robber is “locked” between u and v r u

SLIDE 98

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

8/21

Cops and Robbers meeting SOC Graphs

Result: Strict outer-confluent graphs have cop-number 2 v Using one cop, robber is “locked” between u and v r u

SLIDE 99

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

8/21

Cops and Robbers meeting SOC Graphs

Result: Strict outer-confluent graphs have cop-number 2 v Using one cop, robber is “locked” between u and v r u

SLIDE 100

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

8/21

Cops and Robbers meeting SOC Graphs

Result: Strict outer-confluent graphs have cop-number 2 v Using one cop, robber is “locked” between u and v r How to lock robber to smaller set of vertices? u

SLIDE 101

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

9/21

Moving cops – Case 1

u w v r Result: Strict outer-confluent graphs have cop-number 2 ∃uw-path under which we can lock the robber

SLIDE 102

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

9/21

Moving cops – Case 1

u w v r Result: Strict outer-confluent graphs have cop-number 2 ∃uw-path under which we can lock the robber

SLIDE 103

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

9/21

Moving cops – Case 1

u w v r Result: Strict outer-confluent graphs have cop-number 2 ∃uw-path under which we can lock the robber

SLIDE 104

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

10/21

Moving Cops – Case 2

Result: Strict outer-confluent graphs have cop-number 2 u v r x w ∃xw-path with x, w neighbors of u, v under which we can lock the robber

SLIDE 105

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

10/21

Moving Cops – Case 2

Result: Strict outer-confluent graphs have cop-number 2 u v r x w ∃xw-path with x, w neighbors of u, v under which we can lock the robber

SLIDE 106

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

10/21

Moving Cops – Case 2

Result: Strict outer-confluent graphs have cop-number 2 u v r x w ∃xw-path with x, w neighbors of u, v under which we can lock the robber

SLIDE 107

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

11/21

Moving Cops – Case 3

Result: Strict outer-confluent graphs have cop-number 2 u x v w r y z

SLIDE 108

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

11/21

Moving Cops – Case 3

Result: Strict outer-confluent graphs have cop-number 2 u x v w r y z

SLIDE 109

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

11/21

Moving Cops – Case 3

Result: Strict outer-confluent graphs have cop-number 2 u x v w r y z If no wx-path exists, we find y, z such that we can lock a robber that is inside the red region

SLIDE 110

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

12/21

Catching the Robber

Result: Strict outer-confluent graphs have cop-number 2 Each case makes the set of nodes the robber can use smaller

SLIDE 111

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

12/21

Catching the Robber

Result: Strict outer-confluent graphs have cop-number 2 Each case makes the set of nodes the robber can use smaller We show that one case always applies ⇒ Two cops suffice to catch the roober

SLIDE 112

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

12/21

Catching the Robber

Result: Strict outer-confluent graphs have cop-number 2 Each case makes the set of nodes the robber can use smaller We show that one case always applies ⇒ Two cops suffice to catch the roober

SLIDE 113

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

13/21

Conclusions

One main open question Which graphs have a strict outer-confluent drawing?

SLIDE 114

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

13/21

Conclusions

One main open question Which graphs have a strict outer-confluent drawing? The question remains open Not even good intuition if the problem is in P or NP-hard

SLIDE 115

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

13/21

Conclusions

One main open question Which graphs have a strict outer-confluent drawing? The question remains open Not even good intuition if the problem is in P or NP-hard Many other questions Do strict outer-confluent graphs have bounded cliquewidth? Complexity of other types of confluence? What other graph classes admit (strict) confluent drawings? Confluence, but with some allowed crossings? [Bach et al. 16]

SLIDE 116

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

14/21

SLIDE 117

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

15/21

Our Results

unit interval strict-outerconfluent strict-confluent strict bipartite outerconfluent bipartite permutation ∩ domino-free string

uter-string

bipartite permutation comparability circle trapezoid circular arc circle-trapezoid series-parallel pseudo-split chordal polygon-circle interval-filament co-comparability subtree-filament distance-hereditary ∆-confluent tree-like ∆-strict-outerconfluent

bounded cliquewidth

bipartite permutation

? Cop number: c

3 ≤ c ≤ 15 3 ≤ c ≤ 4 c = 2 c = 2

?

c = 1 c = 1

SLIDE 118

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

15/21

Our Results

unit interval strict-outerconfluent strict-confluent strict bipartite outerconfluent bipartite permutation ∩ domino-free string

uter-string

bipartite permutation comparability circle trapezoid circular arc circle-trapezoid series-parallel pseudo-split chordal polygon-circle interval-filament co-comparability subtree-filament distance-hereditary ∆-confluent tree-like ∆-strict-outerconfluent

bounded cliquewidth

bipartite permutation

? Cop number: c

3 ≤ c ≤ 15 3 ≤ c ≤ 4 c = 2 c = 2

?

c = 1 c = 1

SLIDE 119

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

16/21

Construction of Traces

For each u define trace t(u) (these will be the strings) Each trace starts at u in D Viewed from u, t(u) stays on the left side of the paths Given strict confluent drawing D

SLIDE 120

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

16/21

Construction of Traces

t(u) t(v) t(u) t(v) i i w t(w) w t(w)

For each u define trace t(u) (these will be the strings) Each trace starts at u in D Viewed from u, t(u) stays on the left side of the paths Given strict confluent drawing D

SLIDE 121

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

17/21

Two Problems for Strictness

Domino

SLIDE 122

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

17/21

Two Problems for Strictness

Domino

SLIDE 123

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

17/21

Two Problems for Strictness

Domino

SLIDE 124

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

17/21

Two Problems for Strictness

Domino K3,3

SLIDE 125

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

17/21

Two Problems for Strictness

Domino K3,3

SLIDE 126

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

17/21

Two Problems for Strictness

Domino K3,3

SLIDE 127

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

17/21

Two Problems for Strictness

Domino K3,3

SLIDE 128

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

17/21

Two Problems for Strictness

Domino K3,3

SLIDE 129

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

17/21

Two Problems for Strictness

Domino K3,3

SLIDE 130

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

17/21

Two Problems for Strictness

Domino K3,3

SLIDE 131

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

17/21

Two Problems for Strictness

Domino K3,3 Twisted domino Alternating K3,3

SLIDE 132

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

18/21

SOC Drawings of Bipartite Permutation Graphs

Bipartite Permutation graphs (BP) are graphs that are

SLIDE 133

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

18/21

SOC Drawings of Bipartite Permutation Graphs

Bipartite Permutation graphs (BP) are graphs that are Bipartite

SLIDE 134

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

18/21

SOC Drawings of Bipartite Permutation Graphs

Bipartite Permutation graphs (BP) are graphs that are Bipartite Permutation

SLIDE 135

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

18/21

SOC Drawings of Bipartite Permutation Graphs

Bipartite Permutation graphs (BP) are graphs that are Bipartite Permutation Better characerization for us: A B

SLIDE 136

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

18/21

SOC Drawings of Bipartite Permutation Graphs

Bipartite Permutation graphs (BP) are graphs that are Bipartite Permutation Better characerization for us: A B ⇒ Confluent drawing easily possible (Formal proof [Hui et al. 09]) Domino graph is a Bipartite Permutation graph ⇒ strictness?

SLIDE 137

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

19/21

BP Without Dominos

BP without dominos have soc drawings

SLIDE 138

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

19/21

BP Without Dominos

BP without dominos have soc drawings Draw given graph with algorithm by Hui et al.

SLIDE 139

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

19/21

BP Without Dominos

BP without dominos have soc drawings Draw given graph with algorithm by Hui et al. Observations: K3,3 is never alternating Dominos might be twisted

SLIDE 140

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

19/21

BP Without Dominos

BP without dominos have soc drawings Draw given graph with algorithm by Hui et al. Observations: K3,3 is never alternating Dominos might be twisted We can always untwist dominos u1 u2 u3 v1 v2 v3

SLIDE 141

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

19/21

BP Without Dominos

BP without dominos have soc drawings Draw given graph with algorithm by Hui et al. Observations: K3,3 is never alternating Dominos might be twisted We can always untwist dominos u1 u2 u3 v1 v2 v3 u1 u2 u3 v1 v2 v3

SLIDE 142

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

20/21

Bipartite Strict Confluent Drawings

Drawings such that u1 u2 u3 v1 v2 v3

SLIDE 143

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

20/21

Bipartite Strict Confluent Drawings

Drawings such that u1 u2 u3 v1 v2 v3 Vertices drawn in bipartite “manner”

SLIDE 144

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

20/21

Bipartite Strict Confluent Drawings

Drawings such that u1 u2 u3 v1 v2 v3 Vertices drawn in bipartite “manner” Edges drawn as strict confluent network between them

SLIDE 145

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

20/21

Bipartite Strict Confluent Drawings

Drawings such that u1 u2 u3 v1 v2 v3 Vertices drawn in bipartite “manner” Edges drawn as strict confluent network between them Slide before Bipartite permutation graphs without domino have such drawings

SLIDE 146

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

20/21

Bipartite Strict Confluent Drawings

Drawings such that u1 u2 u3 v1 v2 v3 Vertices drawn in bipartite “manner” Edges drawn as strict confluent network between them Slide before Such drawings are bipartite permutation graphs without dominos: Twisted domino is only order admitting bipartite confluent drawing But twisted domino can not be drawn strict outer-confluent Bipartite permutation graphs without domino have such drawings

SLIDE 147

Henry F¨

rster, Robert Ganian, Fabian Klute, Martin N¨
llenburg · On SOC Graphs

21/21