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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