Between 1-planar and quasiplanar graphs:
Fanplanarity
Michael Kaufmann and Torsten Ueckerdt
fan-crossing
Motivation
Topological Graphs Defined by Forbidden Intersection Patterns
config. I
config. II
planar
3-quasiplanar planar
2-fan-crossing free
fan-planar
≤ 3n − 6 edges
≤ 7n + O(1) edges
≤ 4n − 8 edges
≤ 5n − 10 edges
Motivation
Topological Graphs Defined by Forbidden Intersection Patterns
config. I
config. II
planar
3-quasiplanar planar
2-fan-crossing free
fan-planar
≤ 3n − 6 edges
≤ 7n + O(1) edges
≤ 4n − 8 edges
≤ 5n − 10 edges
Theorem
Every simple topological graph G on n ≥ 3 vertices without
configuration I and II and without homeomorphic parallel edges,
has at most 5n − 10 edges and this bound is tight.
Motivation
Topological Graphs Defined by Forbidden Intersection Patterns
config. I
config. II
planar
3-quasiplanar planar
2-fan-crossing free
fan-planar
≤ 3n − 6 edges
≤ 7n + O(1) edges
≤ 4n − 8 edges
≤ 5n − 10 edges
Theorem
Every simple topological graph G on n ≥ 3 vertices without
configuration I and II and without homeomorphic parallel edges,
has at most 5n − 10 edges and this bound is tight.
Corollary
For geometric graphs, it is sufficient to forbid configuration I.
Some maximum fan-planar graphs
Some maximum fan-planar graphs
Some maximum fan-planar graphs
Some maximal fan-planar graphs
Some maximal fan-planar graphs
Open problems
1
What is the maximum number of edges in a simple topological
graph G satisfying forbidden configuration I, i.e., when
configuration II is allowed?
Open problems
1
2
What is the maximum number of edges in a simple topological
graph G satisfying forbidden configuration I, i.e., when
configuration II is allowed?
Can one prove a better bound than 5n − 10 when G is given with
a straight line embedding?
Open problems
1
2
3
What is the maximum number of edges in a simple topological
graph G satisfying forbidden configuration I, i.e., when
configuration II is allowed?
Can one prove a better bound than 5n − 10 when G is given with
a straight line embedding?
How many edges has an n-vertex edge-maximal fan-planar graph
at least?
Open problems
1
2
3
4
What is the maximum number of edges in a simple topological
graph G satisfying forbidden configuration I, i.e., when
configuration II is allowed?
Can one prove a better bound than 5n − 10 when G is given with
a straight line embedding?
How many edges has an n-vertex edge-maximal fan-planar graph
at least?
For k ≥ 1 what is the largest number of edges in an n-vertex
topological graph where no edge is crossed by more than k
independent edges ?
Pach et al.: Grids in topological graphs
Open problems
1
2
3
4
What is the maximum number of edges in a simple topological
graph G satisfying forbidden configuration I, i.e., when
configuration II is allowed?
Can one prove a better bound than 5n − 10 when G is given with
a straight line embedding?
How many edges has an n-vertex edge-maximal fan-planar graph
at least?
For k ≥ 1 what is the largest number of edges in an n-vertex
topological graph where no edge is crossed by more than k
independent edges ?
Pach et al.: Grids in topological graphs
5
Are fan-planar graphs weak bar-1-visible ?
Open problems
1
2
3
4
What is the maximum number of edges in a simple topological
graph G satisfying forbidden configuration I, i.e., when
configuration II is allowed?
Can one prove a better bound than 5n − 10 when G is given with
a straight line embedding?
How many edges has an n-vertex edge-maximal fan-planar graph
at least?
For k ≥ 1 what is the largest number of edges in an n-vertex
topological graph where no edge is crossed by more than k
independent edges ?
Pach et al.: Grids in topological graphs
5
6
Are fan-planar graphs weak bar-1-visible ?
Explore subclasses of fan-planar graphs.
Open problems
1
2
3
4
What is the maximum number of edges in a simple topological
graph G satisfying forbidden configuration I, i.e., when
configuration II is allowed?
Can one prove a better bound than 5n − 10 when G is given with
a straight line embedding?
How many edges has an n-vertex edge-maximal fan-planar graph
at least?
For k ≥ 1 what is the largest number of edges in an n-vertex
topological graph where no edge is crossed by more than k
independent edges ?
Pach et al.: Grids in topological graphs
5
6
7
Are fan-planar graphs weak bar-1-visible ?
Explore subclasses of fan-planar graphs.
Characterize fan-planar graphs: Forbidden subgraphs ? What
about recognition ?