Picking Planar Edges; or,Drawing a Graph with a

Planar Subgraph

Marcus SchaeferDePaul UniversityGD’14 Würzburg

Speaker: Carsten Gutwenger

Partial Planarity

“If you're given a graph in which some edges are allowed to participate in crossings while others must remain uncrossed, how can you draw it, respecting these constraints?”

Partial Planarity: Examples

crossing-free

crossings allowed

Results by Angelini et al, 13

Partial Planarity: poly-line drawing of (wlog straight-line)• always possible if is spanning tree (with 3 bends)

Geometric Partial Planarity: straight-line drawing (no bends)• always possible if is spanning spider or caterpillar • not always possible if is spanning tree

crossing free subgraph

additional edges

Our ResultsTheorem

Partial planarity is solvable in polynomial time.

Theorem Geometric partial 1-planarity is as hard as, the existential theory of the real numbers.

NP⊆∃ℝ⊆PSPACE

edge is 1-planar if it has at most one crossing

Planar?

Yes!

Theorem (Hanani-Tutte)If graph has drawing in which every two independent edges crossan even number of times, then graph is planar.

Algebraic Hanani-Tutte (Wu, Tutte)

planar ↔ there is a plane drawing of

given any drawing of there are in in , in , so that

for all pairs of independent edges in

↔

Polynomial time planarity algorithm,

e

h(e)

t(e)

Partial Planarity, Algebraically is partial planar

given any drawing of there are in in , in , so that

for all pairs of independent edges with

↔

Polynomial time planarity algorithm,

e

h(e)

t(e)

Missing Ingredient

From Removing Independently Even Crossings (Pelsmajer, Schaefer, Štefankovič, 09)

crossing free subgraph

additional edges

Our ResultsTheorem

Partial planarity is solvable in polynomial time.

Theorem Geometric partial 1-planarity is as hard as, the existential theory of the real numbers.

NP⊆∃ℝ⊆PSPACE

edge is 1-planar if it has at most one crossing

Existential Theory of the Real Numbers

∃𝑥 , 𝑦 ,𝑧 :𝑥2=𝑥∧ 𝑦2=𝑦∧𝑥<𝑦∧𝑧 2=𝑦+𝑦E.g.

Stretchability of Pseudoline Arrangements

Not stretchable (Pappus’ Configuration)

Pseudoline arrangement Equivalent line arrangement

Our ResultsTheorem

Partial planarity is solvable in polynomial time.

Theorem Geometric partial 1-planarity is as hard as, the existential theory of the real numbers.

NP⊆∃ℝ⊆PSPACE

edge is 1-planar if it has at most one crossing

Weak Realizability

is weakly realizable:only pairs of edges in may cross

Problem Complexity

Planarity Linear Time

complete, spanning Trivially True O(1)

complete, not spanning ? ?

Weak Realizability

Problem Complexity

Planarity Linear Time

complete, spanning Trivially True O(1)

complete, not spanning Partial Planarity P

complete, bipartite simultaneous planarity ( ?

complete, n-partite sunflower case of NP-complete

? ? P

is weakly realizable:only pairs of edges in may cross

Excluded Minors: All?

crossing-free

crossings allowed

Operations• Delete vertex, edge• Contract / edge• Turn / into / edge

Excluded Minors: All?

crossing-free

crossings allowed

Operations• Delete vertex, edge• Contract / edge• Turn / into / egeBojan M

ohar

ConjectureGeometric partial planarity is -complete.

Theorem Geometric 1-planarity is NP-complete.

but

Thank You