1. Picking Planar Edges; or, Drawing a Graph with a Planar Subgraph Marcus Schaefer DePaul University GD’14 Würzburg Speaker: Carsten Gutwenger

3. 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?”

4. Partial Planarity: Examples crossing-free crossings allowed

5. Results by Angelini et al, 13

6. Our Results Theorem Partial planarity is solvable in polynomial time. Theorem Partial planarity is solvable in polynomial time. edge is 1-planar if it has at most one crossing

7. Planar?

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

9. Algebraic Hanani-Tutte (Wu, Tutte) ↔ e h(e) t(e)

10. Partial Planarity, Algebraically ↔ e h(e) t(e)

11. Missing Ingredient From Removing Independently Even Crossings (Pelsmajer, Schaefer, Štefankovič, 09)

12. Our Results Theorem Partial planarity is solvable in polynomial time. Theorem Partial planarity is solvable in polynomial time. edge is 1-planar if it has at most one crossing

13. Existential Theory of the Real Numbers E.g.

14. Stretchability of Pseudoline Arrangements Not stretchable (Pappus’ Configuration) Pseudoline arrangementEquivalent line arrangement

16. Our Results Theorem Partial planarity is solvable in polynomial time. Theorem Partial planarity is solvable in polynomial time. edge is 1-planar if it has at most one crossing

17. Weak Realizability ProblemComplexity PlanarityLinear Time complete, spanningTrivially TrueO(1) complete, not spanning??

18. Weak Realizability ProblemComplexity PlanarityLinear Time complete, spanningTrivially TrueO(1) complete, not spanningPartial PlanarityP complete, bipartite? complete, n-partiteNP-complete ??P

19. Excluded Minors: All? crossing-free crossings allowed Operations Delete vertex, edge Contract / edge Turn / into / edge

20. Excluded Minors: All? crossing-free crossings allowed Operations Delete vertex, edge Contract / edge Turn / into / ege Bojan Mohar

21. Theorem Geometric 1-planarity is NP-complete. Theorem Geometric 1-planarity is NP-complete. but