from a sat-solving perspective the book embedding problem
TRANSCRIPT
Michalis Bekos, Michael Kaufmann, Christian Zielke
Universitat Tubingen, Germany
The Book Embedding Problemfrom a SAT-Solving Perspective
[GD 2015]
The Book Embedding problem
Vertices V ordered, on a spine→
Edges E assigned to p pages→
Pages half planes, delimited by spineplanar
→
The Book Embedding problem
Vertices V ordered, on a spine→
Edges E assigned to p pages→
Pages half planes, delimited by spineplanar
→
goal: minimize p - book thickness
The Book Embedding problem
Vertices V ordered, on a spine→
Edges E assigned to p pages→
Pages half planes, delimited by spineplanar
→
1
2
3 4 5 6 7
8 9
10 11
goal: minimize p - book thickness
The Book Embedding problem
Vertices V ordered, on a spine→
Edges E assigned to p pages→
Pages half planes, delimited by spineplanar
→
1
2
3 4 5 6 7
8 9
10 11
1 7 6 11 5 10 2 3 4 8 9
goal: minimize p - book thickness
The Book Embedding problem
Known results:
[Bernhard & Kainen, 1979]
Vertices V ordered, on a spine→
Edges E assigned to p pages→
Pages half planes, delimited by spineplanar
→
1
2
3 4 5 6 7
8 9
10 11
1 7 6 11 5 10 2 3 4 8 9
goal: minimize p - book thickness
bt(G ) = 1⇔ G is outerplanar, bt(G ) = 2⇔ G is subhamiltonian
The Book Embedding problem
all planar graphs can be embedded on 4 pages
Known results:
[Yannakakis, 1989]
[Bernhard & Kainen, 1979]
Vertices V ordered, on a spine→
Edges E assigned to p pages→
Pages half planes, delimited by spineplanar
→
(no graph needing 4 pages is known)
1
2
3 4 5 6 7
8 9
10 11
1 7 6 11 5 10 2 3 4 8 9
goal: minimize p - book thickness
bt(G ) = 1⇔ G is outerplanar, bt(G ) = 2⇔ G is subhamiltonian
The Book Embedding problem
all planar graphs can be embedded on 4 pages
Known results:
[Yannakakis, 1989]
[Bernhard & Kainen, 1979]
1-planar graphs have constant book thickness
[Bekos et al., 2015]
Vertices V ordered, on a spine→
Edges E assigned to p pages→
Pages half planes, delimited by spineplanar
→
(no graph needing 4 pages is known)
(39)
1
2
3 4 5 6 7
8 9
10 11
1 7 6 11 5 10 2 3 4 8 9
goal: minimize p - book thickness
bt(G ) = 1⇔ G is outerplanar, bt(G ) = 2⇔ G is subhamiltonian
SAT solving
Boolean variables
clauses
”sun is shining”, ”road is wet”, α, β, ...
conjunctive normal form (α ∨ ¬β) ∧ (... ∨ ... ∨ ...) ∧ ...
SAT solving
Boolean variables
clauses
”sun is shining”, ”road is wet”, α, β, ...
conjunctive normal form (α ∨ ¬β) ∧ (... ∨ ... ∨ ...) ∧ ...formula
SAT solving
logical operations
→ implications: ”if ... then ...”
↔ equivalence: ”if and only if”
Boolean variables
clauses
”sun is shining”, ”road is wet”, α, β, ...
conjunctive normal form (α ∨ ¬β) ∧ (... ∨ ... ∨ ...) ∧ ...formula
SAT solving
logical operations
→ implications: ”if ... then ...”
↔ equivalence: ”if and only if”
”forbid”particular settings
Boolean variables
clauses
”sun is shining”, ”road is wet”, α, β, ...
conjunctive normal form (α ∨ ¬β) ∧ (... ∨ ... ∨ ...) ∧ ...
(α ∨ ¬β) forbids ¬α,β
formula
SAT solving
logical operations
→ implications: ”if ... then ...”
↔ equivalence: ”if and only if”
”forbid”particular settings
Boolean variables
clauses
”sun is shining”, ”road is wet”, α, β, ...
conjunctive normal form (α ∨ ¬β) ∧ (... ∨ ... ∨ ...) ∧ ...
(α ∨ ¬β) forbids ¬α,β
SAT? assignment to variables that satisfies all clauses
formula
SAT solving
logical operations
→ implications: ”if ... then ...”
↔ equivalence: ”if and only if”
”forbid”particular settings
Boolean variables
clauses
”sun is shining”, ”road is wet”, α, β, ...
conjunctive normal form (α ∨ ¬β) ∧ (... ∨ ... ∨ ...) ∧ ...
(α ∨ ¬β) forbids ¬α,β
SAT? assignment to variables that satisfies all clauses
formula
use SAT solver as black box / oracle
SAT solving
logical operations
→ implications: ”if ... then ...”
↔ equivalence: ”if and only if”
”forbid”particular settings
Boolean variables
clauses
”sun is shining”, ”road is wet”, α, β, ...
conjunctive normal form (α ∨ ¬β) ∧ (... ∨ ... ∨ ...) ∧ ...
(α ∨ ¬β) forbids ¬α,β
formula
Applications of SAT:
[Zeranski & Chimani, 2012]
Upward planarity via SAT
SAT solving
logical operations
→ implications: ”if ... then ...”
↔ equivalence: ”if and only if”
”forbid”particular settings
Boolean variables
clauses
”sun is shining”, ”road is wet”, α, β, ...
conjunctive normal form (α ∨ ¬β) ∧ (... ∨ ... ∨ ...) ∧ ...
(α ∨ ¬β) forbids ¬α,β
formula
Applications of SAT:
[Biedl et al, 2013]
SAT for grid-based graph problems (pathwidth, vis. representation, ...)
[Zeranski & Chimani, 2012]
Upward planarity via SAT
SAT formulation
General idea:
1. ensure a proper order of the vertices on the spine1.
build formula F(G , p) via
SAT formulation
General idea:
1. ensure a proper order of the vertices on the spine1.
vi
build formula F(G , p) via
SAT formulation
General idea:
1. ensure a proper order of the vertices on the spine1.
vi vj
σ(vi , vj) : vi is left of vj on spine
Variables:
build formula F(G , p) via
SAT formulation
General idea:
1. ensure a proper order of the vertices on the spine1.
vi vj
σ(vi , vj) : vi is left of vj on spine
Variables:
Rules:
build formula F(G , p) via
SAT formulation
General idea:
1. ensure a proper order of the vertices on the spine1.
vi vj
σ(vi , vj) : vi is left of vj on spine
Variables:
Rules:
Antisymmetry: σ(vi , vj)↔ ¬σ(vj , vi )
build formula F(G , p) via
SAT formulation
General idea:
1. ensure a proper order of the vertices on the spine1.
vi vjvj
σ(vi , vj) : vi is left of vj on spine
Variables:
Rules:
Antisymmetry: σ(vi , vj)↔ ¬σ(vj , vi )
build formula F(G , p) via
SAT formulation
General idea:
1. ensure a proper order of the vertices on the spine1.
vi vj
σ(vi , vj) : vi is left of vj on spine
Variables:
Rules:
Antisymmetry: σ(vi , vj)↔ ¬σ(vj , vi )
Transitivity: σ(vi , vj) ∧ σ(vj , vk)→ σ(vi , vk)
build formula F(G , p) via
SAT formulation
General idea:
1. ensure a proper order of the vertices on the spine1.
vi vj vk
σ(vi , vj) : vi is left of vj on spine
Variables:
Rules:
Antisymmetry: σ(vi , vj)↔ ¬σ(vj , vi )
Transitivity: σ(vi , vj) ∧ σ(vj , vk)→ σ(vi , vk)
build formula F(G , p) via
SAT formulation
General idea:
1. ensure a proper order of the vertices on the spine1.
vi vj
X
σ(vi , vj)
build formula F(G , p) via
SAT formulation
General idea:
1.
2.
ensure a proper order of the vertices on the spine
assure that every edge is assigned to one of p pages
1.
2.
vi vj
X
σ(vi , vj)
Variables:φp(ei ) : edge ei is assigned to page p
build formula F(G , p) via
SAT formulation
General idea:
1.
2.
ensure a proper order of the vertices on the spine
assure that every edge is assigned to one of p pages
1.
2.
vi vj
X
σ(vi , vj)
Variables:φp(ei ) : edge ei is assigned to page p
Rules:
build formula F(G , p) via
SAT formulation
General idea:
1.
2.
ensure a proper order of the vertices on the spine
assure that every edge is assigned to one of p pages
1.
2.
vi vj
X
σ(vi , vj)
Variables:φp(ei ) : edge ei is assigned to page p
Rules:≥ 1 page: φ1(ei ) ∨ φ2(ei ) ∨ . . . ∨ φp(ei )
build formula F(G , p) via
SAT formulation
General idea:
1.
2.
ensure a proper order of the vertices on the spine
assure that every edge is assigned to one of p pages
1.
2.
vi vj
X
σ(vi , vj)
Variables:φp(ei ) : edge ei is assigned to page p
Rules:≥ 1 page: φ1(ei ) ∨ φ2(ei ) ∨ . . . ∨ φp(ei )
χ(ei , ej) : edges ei and ej are assigned to same page
build formula F(G , p) via
SAT formulation
General idea:
1.
2.
ensure a proper order of the vertices on the spine
assure that every edge is assigned to one of p pages
1.
2.
vi vj
X
σ(vi , vj)
Variables:φp(ei ) : edge ei is assigned to page p
Rules:≥ 1 page: φ1(ei ) ∨ φ2(ei ) ∨ . . . ∨ φp(ei )
χ(ei , ej) : edges ei and ej are assigned to same page
same page: (φk(ei ) ∧ φk(ej))→ χ(ei , ej)
build formula F(G , p) via
SAT formulation
General idea:
1.
2.
ensure a proper order of the vertices on the spine
assure that every edge is assigned to one of p pages
1.
2.
vi vj
X
X
σ(vi , vj)
χ(ei , ej)ei ej
build formula F(G , p) via
SAT formulation
General idea:
1.
2.
3.
ensure a proper order of the vertices on the spine
assure that every edge is assigned to one of p pages
1.
2.
forbid crossings on each page3.
vi vj
X
X
σ(vi , vj)
χ(ei , ej)ei ej
build formula F(G , p) via
SAT formulation
General idea:
1.
2.
3.
ensure a proper order of the vertices on the spine
assure that every edge is assigned to one of p pages
1.
2.
forbid crossings on each page3.
vi vj
X
X
σ(vi , vj)
χ(ei , ej)ei ej
forbid χ(ei , ej) together withei ej
build formula F(G , p) via
SAT formulation
General idea:
1.
2.
3.
ensure a proper order of the vertices on the spine
assure that every edge is assigned to one of p pages
1.
2.
forbid crossings on each page3.
vi vj
X
X
σ(vi , vj)
χ(ei , ej)ei ej
vi vjvk vl
forbid χ(ei , ej) together withei ej
σ(vi , vk),σ(vk , vj),σ(vj , vl)
build formula F(G , p) via
SAT formulation
General idea:
1.
2.
3.
ensure a proper order of the vertices on the spine
assure that every edge is assigned to one of p pages
1.
2.
forbid crossings on each page3.
vi vj
X
X
X
σ(vi , vj)
χ(ei , ej)ei ej
vi vjvk vl
forbid χ(ei , ej) together withei ej
σ(vi , vk),σ(vk , vj),σ(vj , vl)(for every pair of edges 8 forbidden configurations)
build formula F(G , p) via
SAT formulation
General idea:
1.
2.
3.
ensure a proper order of the vertices on the spine
assure that every edge is assigned to one of p pages
1.
2.
forbid crossings on each page3.
vi vj
X
X
X
σ(vi , vj)
χ(ei , ej)ei ej
build formula F(G , p) via
solve optimization problem: F(G , k − 1) is UNSAT, F(G , k) is SAT
Experiments
Setup:
all Rome and North graphs (taken from www.graphdrawing.org)
Rome graphs
planar: 3281
nonplanar: 8253(graphs are very sparse: 0.069)
Timeout: 1200 sec
Experiments
Setup:
all Rome and North graphs (taken from www.graphdrawing.org)
Rome graphs
planar: 3281 all fit in 2 pages
nonplanar: 8253 all fit in 3 pages(graphs are very sparse: 0.069)
Timeout: 1200 sec
Experiments
Setup:
all Rome and North graphs (taken from www.graphdrawing.org)
Rome graphs
planar: 3281 all fit in 2 pages
nonplanar: 8253 all fit in 3 pages(graphs are very sparse: 0.069)
North graphs
planar: 854
nonplanar: 423
Timeout: 1200 sec
(graphs are denser: 0.13)
Experiments
Setup:
all Rome and North graphs (taken from www.graphdrawing.org)
Rome graphs
planar: 3281 all fit in 2 pages
nonplanar: 8253 all fit in 3 pages(graphs are very sparse: 0.069)
North graphs
planar: 854 all fit in 2 pages
nonplanar: 423 only 344 were solved completely
Timeout: 1200 sec
(graphs are denser: 0.13)
Experiments
Setup:
all Rome and North graphs (taken from www.graphdrawing.org)
Rome graphs
planar: 3281 all fit in 2 pages
nonplanar: 8253 all fit in 3 pages(graphs are very sparse: 0.069)
North graphs
planar: 854 all fit in 2 pages
nonplanar: 423 only 344 were solved completely
Timeout: 1200 sec
(graphs are denser: 0.13)(some graphs have high book thickness)
Experiments
Hypothesis:
There is a (maximal) planar graph whose book thickness is 4.
Setup:
several maximal planar graphs with 500-700 vertices
Experiments
Hypothesis:
There is a (maximal) planar graph whose book thickness is 4.
Setup:
several maximal planar graphs with 500-700 vertices
Construction:
1. triangulated graph as base (”skeleton”)1. (not necessarily non-Hamiltonian)
Experiments
Hypothesis:
There is a (maximal) planar graph whose book thickness is 4.
Setup:
several maximal planar graphs with 500-700 vertices
Construction:
1.
2.
triangulated graph as base (”skeleton”)
augmenting the faces of the base graph by a combination of
1.
2.
(not necessarily non-Hamiltonian)
Stellation Octahedron Creation Graph Insertion
Experiments
Hypothesis:
There is a (maximal) planar graph whose book thickness is 4.
Setup:
several maximal planar graphs with 500-700 vertices
Construction:
1.
2.
triangulated graph as base (”skeleton”)
augmenting the faces of the base graph by a combination of
1.
2.
(not necessarily non-Hamiltonian)
Stellation Octahedron Creation Graph Insertion
Experiments
Hypothesis:
There is a (maximal) planar graph whose book thickness is 4.
Setup:
several maximal planar graphs with 500-700 vertices
Construction:
1.
2.
triangulated graph as base (”skeleton”)
augmenting the faces of the base graph by a combination of
1.
2.
(not necessarily non-Hamiltonian)
Stellation Octahedron Creation Graph Insertion
Experiments
Hypothesis:
There is a (maximal) planar graph whose book thickness is 4.
Setup:
several maximal planar graphs with 500-700 vertices
Construction:
1.
2.
triangulated graph as base (”skeleton”)
augmenting the faces of the base graph by a combination of
1.
2.
(not necessarily non-Hamiltonian)
Stellation Octahedron Creation Graph Insertion
Experiments
Hypothesis:
There is a (maximal) planar graph whose book thickness is 4.
Setup:
several maximal planar graphs with 500-700 vertices
Construction:
1.
2.
triangulated graph as base (”skeleton”)
augmenting the faces of the base graph by a combination of
1.
2.
(not necessarily non-Hamiltonian)
Stellation Octahedron Creation Graph Insertion
Experiments
Hypothesis:
There is a (maximal) planar graph whose book thickness is 4.
Setup:
several maximal planar graphs with 500-700 vertices
Construction:
1.
2.
triangulated graph as base (”skeleton”)
augmenting the faces of the base graph by a combination of
1.
2.
(not necessarily non-Hamiltonian)
Stellation Octahedron Creation Graph Insertion
Experiments
Hypothesis:
There is a (maximal) planar graph whose book thickness is 4.
Setup:
several maximal planar graphs with 500-700 vertices
Construction:
1.
2.
triangulated graph as base (”skeleton”)
augmenting the faces of the base graph by a combination of
1.
2.
(not necessarily non-Hamiltonian)
Stellation Octahedron Creation Graph Insertion
Experiments
Hypothesis:
There is a (maximal) planar graph whose book thickness is 4.
Setup:
several maximal planar graphs with 500-700 vertices
Construction:
1.
2.
triangulated graph as base (”skeleton”)
augmenting the faces of the base graph by a combination of
1.
2.
(not necessarily non-Hamiltonian)
Stellation Octahedron Creation Graph Insertion
Experiments
Hypothesis:
There is a (maximal) planar graph whose book thickness is 4.
Setup:
several maximal planar graphs with 500-700 vertices
Construction:
1.
2.
triangulated graph as base (”skeleton”)
augmenting the faces of the base graph by a combination of
1.
2.
(not necessarily non-Hamiltonian)
Stellation Octahedron Creation Graph Insertion
(all 3 page embeddable)
Experiments
- Idea: possibly overcome the limit of ≈ 700 vertices
Hypotheses:
There is a maximal planar graph Ga that has at least one face fa whoseedges cannot be on the same page in any book embedding on 3 pages.
Experiments
- Idea: possibly overcome the limit of ≈ 700 vertices
Hypotheses:
There is a maximal planar graph Ga that has at least one face fa whoseedges cannot be on the same page in any book embedding on 3 pages.
test F(Ga, 3) ∪ {(χ(ei , ej) ∧ χ(ei , ek)}∀fa = (ei , ej , ek) ∈ Ga
Experiments
- Idea: possibly overcome the limit of ≈ 700 vertices
Hypotheses:
There is a maximal planar graph Ga that has at least one face fa whoseedges cannot be on the same page in any book embedding on 3 pages.
There is a maximal planar graph Gc that has at least one face fc whoseedges are on the same page in any book embedding on 3 pages.
test F(Ga, 3) ∪ {(χ(ei , ej) ∧ χ(ei , ek)}∀fa = (ei , ej , ek) ∈ Ga
Experiments
- Idea: possibly overcome the limit of ≈ 700 vertices
Hypotheses:
There is a maximal planar graph Ga that has at least one face fa whoseedges cannot be on the same page in any book embedding on 3 pages.
There is a maximal planar graph Gc that has at least one face fc whoseedges are on the same page in any book embedding on 3 pages.
∀fc = (ei , ej , ek) ∈ Gc
test F(Gc , 3) ∪ (¬χ(ei , ej) ∨ ¬χ(ei , ek))
test F(Ga, 3) ∪ {(χ(ei , ej) ∧ χ(ei , ek)}∀fa = (ei , ej , ek) ∈ Ga
Experiments
- Idea: possibly overcome the limit of ≈ 700 vertices
Hypotheses:
There is a maximal planar graph Ga that has at least one face fa whoseedges cannot be on the same page in any book embedding on 3 pages.
There is a maximal planar graph Gc that has at least one face fc whoseedges are on the same page in any book embedding on 3 pages.
∀fc = (ei , ej , ek) ∈ Gc
test F(Gc , 3) ∪ (¬χ(ei , ej) ∨ ¬χ(ei , ek))
test F(Ga, 3) ∪ {(χ(ei , ej) ∧ χ(ei , ek)}∀fa = (ei , ej , ek) ∈ Ga
Gc
Experiments
- Idea: possibly overcome the limit of ≈ 700 vertices
Hypotheses:
There is a maximal planar graph Ga that has at least one face fa whoseedges cannot be on the same page in any book embedding on 3 pages.
There is a maximal planar graph Gc that has at least one face fc whoseedges are on the same page in any book embedding on 3 pages.
∀fc = (ei , ej , ek) ∈ Gc
test F(Gc , 3) ∪ (¬χ(ei , ej) ∨ ¬χ(ei , ek))
test F(Ga, 3) ∪ {(χ(ei , ej) ∧ χ(ei , ek)}∀fa = (ei , ej , ek) ∈ Ga
Gc Gafa is outer face
Experiments
- Idea: possibly overcome the limit of ≈ 700 vertices
Hypotheses:
There is a maximal planar graph Ga that has at least one face fa whoseedges cannot be on the same page in any book embedding on 3 pages.
There is a maximal planar graph Gc that has at least one face fc whoseedges are on the same page in any book embedding on 3 pages.
∀fc = (ei , ej , ek) ∈ Gc
test F(Gc , 3) ∪ (¬χ(ei , ej) ∨ ¬χ(ei , ek))
test F(Ga, 3) ∪ {(χ(ei , ej) ∧ χ(ei , ek)}∀fa = (ei , ej , ek) ∈ Ga
GcGa
Ga
Experiments
- Idea: possibly overcome the limit of ≈ 700 vertices
Hypotheses:
There is a maximal planar graph Ga that has at least one face fa whoseedges cannot be on the same page in any book embedding on 3 pages.
There is a maximal planar graph Gc that has at least one face fc whoseedges are on the same page in any book embedding on 3 pages.
∀fc = (ei , ej , ek) ∈ Gc
test F(Gc , 3) ∪ (¬χ(ei , ej) ∨ ¬χ(ei , ek))
test F(Ga, 3) ∪ {(χ(ei , ej) ∧ χ(ei , ek)}∀fa = (ei , ej , ek) ∈ Ga
GcGa
Ga tested ≈ 284,000 graphswith 60 to 125 vertices→ 0 confirmed Hypotheses
Experiments
Hypothesis:
There is a 1-planar graph whose book thickness is (at least) 4.
1
2 3
4
5
6 7
8
Experiments
Hypothesis:
There is a 1-planar graph whose book thickness is (at least) 4.
1
2 3
4
5
6 7
8
1 2 345 67 8
Experiments
Hypothesis:
Setup:
all 2,098,675 planar quadrangulations with 25 vertices
There is a 1-planar graph whose book thickness is (at least) 5.
triconnected; min degree 3 → augment every face with two crossing edges
Experiments
Hypothesis:
Setup:
all 2,098,675 planar quadrangulations with 25 vertices (all 4 page embeddable)
There is a 1-planar graph whose book thickness is (at least) 5.
triconnected; min degree 3 → augment every face with two crossing edges
Experiments
Hypothesis:
Setup:
all 2,098,675 planar quadrangulations with 25 vertices (all 4 page embeddable)
There is a 1-planar graph whose book thickness is (at least) 5.
triconnected; min degree 3 → augment every face with two crossing edges
Experiments
Hypothesis:
Setup:
all 2,098,675 planar quadrangulations with 25 vertices (all 4 page embeddable)
There is a 1-planar graph whose book thickness is (at least) 5.
triconnected; min degree 3 → augment every face with two crossing edges
8312 random optimal 1-planar graphs with 50− 155 vertices(all 4 page embeddable)
Experiments
Hypothesis:
Setup:
all 2,098,675 planar quadrangulations with 25 vertices (all 4 page embeddable)
There is a 1-planar graph whose book thickness is (at least) 5.
triconnected; min degree 3 → augment every face with two crossing edges
8312 random optimal 1-planar graphs with 50− 155 vertices(all 4 page embeddable)
Experiments
Hypothesis:
There is a (maximal) planar graph, which cannot be embeddedin a book of 3 pages, if the subgraphs at each page must be acyclic.
Experiments
Hypothesis:
There is a (maximal) planar graph, which cannot be embeddedin a book of 3 pages, if the subgraphs at each page must be acyclic.
(Heath used that for proving book thickness of 3 for planar 3-trees)
Experiments
Hypothesis:
There is a (maximal) planar graph, which cannot be embeddedin a book of 3 pages, if the subgraphs at each page must be acyclic.
Setup:
tested over 15,000 maximal planar graphs with 25 to 80 vertices
(Heath used that for proving book thickness of 3 for planar 3-trees)
Experiments
Hypothesis:
There is a (maximal) planar graph, which cannot be embeddedin a book of 3 pages, if the subgraphs at each page must be acyclic.
Setup:
tested over 15,000 maximal planar graphs with 25 to 80 vertices
70.78 % of graphs were solved ≤ 3 minutes
Timeout: 1200 sec
76.37 % of graphs were solved ≤ 20 minutes
(Heath used that for proving book thickness of 3 for planar 3-trees)
Experiments
Hypothesis:
There is a (maximal) planar graph, which cannot be embeddedin a book of 3 pages, if the subgraphs at each page must be acyclic.
Setup:
tested over 15,000 maximal planar graphs with 25 to 80 vertices
70.78 % of graphs were solved ≤ 3 minutes
Timeout: 1200 sec
76.37 % of graphs were solved ≤ 20 minutes
→ additional constraints to force acyclic subgraphs increase runtime
(Heath used that for proving book thickness of 3 for planar 3-trees)
Experiments
Hypothesis:
There is a (maximal) planar graph, which cannot be embeddedin a book of 3 pages, if the subgraphs at each page must be acyclic.
(weaker)
subgraphs are trees on n − 1 vertices
vertices not spanned by treesbuild a face(w.l.o.g. outer face)
Experiments
Hypothesis:
There is a (maximal) planar graph, which cannot be embeddedin a book of 3 pages, if the subgraphs at each page must be acyclic.
(weaker)
subgraphs are trees on n − 1 vertices
vertices not spanned by treesbuild a face(w.l.o.g. outer face)
1
16
2
3 45
6
7 8
9
15
10
1112 13
14
Experiments
Hypothesis:
There is a (maximal) planar graph, which cannot be embeddedin a book of 3 pages, if the subgraphs at each page must be acyclic.
(weaker)
subgraphs are trees on n − 1 vertices
vertices not spanned by treesbuild a face(w.l.o.g. outer face)
1
16
2
3 45
6
7 8
9
15
10
1112 13
14
→ Schnyder decompositionnot always possible
Open problems
All Hypothesis are unproven!
- 1: (maximal) planar graph that requires 4 pages
- 1a: (maximal) planar graph that requires unicolored face
Open problems
All Hypothesis are unproven!
- 1: (maximal) planar graph that requires 4 pages
- 1a: (maximal) planar graph that requires unicolored face
- 1b: (maximal) planar graph that cannot have a unicolored face
Open problems
All Hypothesis are unproven!
- 1: (maximal) planar graph that requires 4 pages
- 1a: (maximal) planar graph that requires unicolored face
- 1b: (maximal) planar graph that cannot have a unicolored face
- 2: 1-planar graph that requires 5 pages
Open problems
All Hypothesis are unproven!
- 1: (maximal) planar graph that requires 4 pages
- 1a: (maximal) planar graph that requires unicolored face
- 1b: (maximal) planar graph that cannot have a unicolored face
- 2: 1-planar graph that requires 5 pages
- 3: (maximal) planar graph, that requires cyclic subgraphs on pages
Open problems
All Hypothesis are unproven!
- 1: (maximal) planar graph that requires 4 pages
- 1a: (maximal) planar graph that requires unicolored face
- 1b: (maximal) planar graph that cannot have a unicolored face
- 2: 1-planar graph that requires 5 pages
- 3: (maximal) planar graph, that requires cyclic subgraphs on pages
Improve the Encoding!
Open problems
All Hypothesis are unproven!
- 1: (maximal) planar graph that requires 4 pages
- 1a: (maximal) planar graph that requires unicolored face
- 1b: (maximal) planar graph that cannot have a unicolored face
- 2: 1-planar graph that requires 5 pages
- 3: (maximal) planar graph, that requires cyclic subgraphs on pages
Improve the Encoding!
- cope with graphs, that have a high number of vertices and edges
Open problems
All Hypothesis are unproven!
- 1: (maximal) planar graph that requires 4 pages
- 1a: (maximal) planar graph that requires unicolored face
- 1b: (maximal) planar graph that cannot have a unicolored face
- 2: 1-planar graph that requires 5 pages
- 3: (maximal) planar graph, that requires cyclic subgraphs on pages
Improve the Encoding!
- cope with graphs, that have a high number of vertices and edges
- cope with graphs, that have high book thickness
Open problems
All Hypothesis are unproven!
Thanks!!!
- 1: (maximal) planar graph that requires 4 pages
- 1a: (maximal) planar graph that requires unicolored face
- 1b: (maximal) planar graph that cannot have a unicolored face
- 2: 1-planar graph that requires 5 pages
- 3: (maximal) planar graph, that requires cyclic subgraphs on pages
Improve the Encoding!
- cope with graphs, that have a high number of vertices and edges
- cope with graphs, that have high book thickness
Experiments
Hypothesis:
Setup:
all 2,098,675 planar quadrangulations with 25 vertices
Construction:
1. start with cube1.
(all 4 page embeddable)
There is a 1-planar graph whose book thickness is (at least) 5.
triconnected; min degree 3 → augment every face with two crossing edges
8312 random optimal 1-planar graphs with 50− 155 vertices
Experiments
Hypothesis:
Setup:
all 2,098,675 planar quadrangulations with 25 vertices
Construction:
1.
2.
start with cube
apply one of the two operations defined by
1.
2.
(all 4 page embeddable)
There is a 1-planar graph whose book thickness is (at least) 5.
triconnected; min degree 3 → augment every face with two crossing edges
8312 random optimal 1-planar graphs with 50− 155 vertices
[Suzuki, 2006]
Experiments
Hypothesis:
Setup:
all 2,098,675 planar quadrangulations with 25 vertices
Construction:
1.
2.
start with cube
apply one of the two operations defined by
1.
2.
(all 4 page embeddable)
There is a 1-planar graph whose book thickness is (at least) 5.
triconnected; min degree 3 → augment every face with two crossing edges
8312 random optimal 1-planar graphs with 50− 155 vertices
[Suzuki, 2006]
Experiments
Hypothesis:
Setup:
all 2,098,675 planar quadrangulations with 25 vertices
Construction:
1.
2.
start with cube
apply one of the two operations defined by
1.
2.
(all 4 page embeddable)
There is a 1-planar graph whose book thickness is (at least) 5.
triconnected; min degree 3 → augment every face with two crossing edges
8312 random optimal 1-planar graphs with 50− 155 vertices
[Suzuki, 2006]
Experiments
Hypothesis:
Setup:
all 2,098,675 planar quadrangulations with 25 vertices
Construction:
1.
2.
start with cube
apply one of the two operations defined by
1.
2.
(all 4 page embeddable)
There is a 1-planar graph whose book thickness is (at least) 5.
triconnected; min degree 3 → augment every face with two crossing edges
8312 random optimal 1-planar graphs with 50− 155 vertices
[Suzuki, 2006]
(all 4 page embeddable)
Experiments
Hypothesis:
Setup:
all 2,098,675 planar quadrangulations with 25 vertices (all 4 page embeddable)
There is a 1-planar graph whose book thickness is (at least) 5.
triconnected; min degree 3 → augment every face with two crossing edges
8312 random optimal 1-planar graphs with 50− 155 vertices(all 4 page embeddable)