on extending a partial straight-line drawing

1. on extending a partial straight-line drawing maurizio “titto” patrignani third university of rome graph drawing 2005

2. straight-line drawings G(V,E)

3. straight-line drawings of planar graphs • every planar graph admits a planar straight-line drawing [steinitz and rademacher, ‘34], [wagner, ‘36], [fary, ‘48], and [stein, ‘51] • a planar straight-line drawing can be computed in linear time

4. partial drawing extensibility (pde) instance: a planar graph G(V,E) and a mapping between a subset V' of its vertices and a set of distinct points of the plane question: can coordinates be assigned to the vertices in V-V' such that the resulting straight-line drawing of G(V,E) is planar?

5. V-V' free vertices example of partial drawing extensibility V' G(V,E)

6. problem • establish the complexity of the partial drawing extensibility problem • we show that it is NP-complete • problem #9 of [brandenburg, eppstein, goodrich, kobourov, liotta, and mutzel. ’03] • related to problem #10 (drawing with fixed vertex positions)? • problem #10 shown to be NP-hard by [cabello, 2004]

7. 3-satisfiability (3sat) instance: a set of clauses {c1, c2, …, cm} each one having three literals from a set of boolean variables {v1, v2, …, vn} question: can truth values be assigned to the variables so that each clause contains at least one true literal? example of 3sat instance: (v1 v3  v4)  (v1 v2  v5)  (v2 v3  v5) c3 c2 c1

8. planar 3-satisfiability (p3sat) [lichtenstein, ‘82] instance: a set of clauses {c1, c2, …, cm}, each one having three literals from a set of boolean variables {v1, v2, …, vn} and a plane bipartite graph G(VA,VB, E) such that: • nodes in VA correspond to the variables while nodes in VB correspond to the clauses (hence, |VA|=n and |VB|=m) • edges connect clauses to the variables of the literals they contain • G(VA,VB,E) is drawn without intersections on a rectangular grid of polynomial size in such a way that nodes in VA are arranged in a horizontal line that is not crossed by any edge question: can truth values be assigned to the variables so that each clause contains at least one true literal? (v1 v1  v2)  (v1 v2  vn)  (v2 v3  v4)  (v2 v3  v5)  (v2 v4  vn)  (v3 v4  v5)  (v2 v5  vn) …

9. pde-instance construction drawing of G(VA,VB, E) c1 c7 c4 c6 v1 v2 v4 v5 v6 vn c3 c5 c2

10. basic gadget exits false gate true gate

11. variable gadget for variable vi vi vertex ni,

12. clause gadgets c1 c7 c4 c6 v1 v2 v4 v5 v6 vn c3 c5 c2

13. clause gadget for clause ch= (vi vj vk) vertex nh,,

14. free vertices

15. partial drawing extensibility is NP-hard theorem partial drawing extensibility is NP-hard proof • starting from a p3sat instance the pde instance can be constructed in polynomial time • if the p3sat instance admits a solution the corresponding pde instance does • if the pde instance admits a solution the original p3sat instance does

16. connected instance

17. conclusions • we showed that the partial drawing extensibility problem is NP-complete • the drawing method of [tutte, 1963] may be used to show that the problem becomes tractable when the graph is triconnected and the already-placed vertices form convex faces • if the already-placed vertices are assumed to form a simple polygon and the graph is assumed to have all interior faces triangles is the problem simpler?

18. thank you for your attention!