1. Recognizing Strings in NP Marcus Schaefer, Eric Sedgwick, Daniel Štefankovič Presentation by Robert Salazar

2. Definitions: Given a collection of curves C i where i I, the intersection graph is defined as (I, {(i, j) : C i and C j intersect}) The size of a collection of curves is the number of intersection points String graph: A graph isomorphic to the intersection graph of a collection of curves String graph problem asks how string graphs can be recognized. This problem was previously shown to be NP-hard.

3. Definitions Graph Drawing problem – Given graph G = (V, E) and a set – A drawing D is a weak realization of (G, R) if only pairs of edges which are in R are allowed to intersect. These edges do not necessarily intersect. – (G, R) is weakly realizible

4. String graph problem reduction Given G = (V, E), let G’ = Let G is a string graph if and only if (G’, R) is weakly realizible

6. Lemmas Let M be a compact, orientable surface with a boundary. A properly embedded arc γ has both endpoints on the boundary δM and all internal points on the interior of M.

7. Lemmas

8. Theorem Let G = (V, E) be a graph with m edges, such that (G, R) is weakly realizable, and let D be a weak realization of (G, R) with the minimal number of intersections. Then for any edge e G there are less than 2 m intersections for the curve realizing e in D. Let M be a compact, orientable surface with a boundary. A properly embedded arc γ has both endpoints on the boundary δM and all internal points on the interior of M.

9. Proof Let G = (V, E) be a graph. Let M be the surface obtained from the plane by drilling |V| holes. Each hole is labeled by a vertex of G. Let. A set S of properly embedded arcs on M is called a weak realization with holes of (G, R) if for each e = {u, v} E there is a properly embedded arc in S connecting hole u to hole v, and if then the properly embedded arcs e, f are isotopically disjoint.

10. Proof (II) Given a weak realization D, drill small holes in place of the vertices to obtain a weak realization with holes. By Lemma 3.2, there is a weak realization with holes in which forthe properly embedded arcs e, f are disjoint. Contracting the holes yields a weak realization of (G, R)

11. Proposition / Lemma

12. Proof Construct a triangulation T with 3n vertices, using 3 vertices for each boundary component (i.e. hole) T has 9n – 6 edged by Euler’s formula

13. Proof (II) Consider: Weak realization problem – Graph H such that and There are edges to all vertices of T which lie on hole v. – Pairs P of edges that may intersect are All pairs in R For every edge select an edge ; ev may intersect with edges in E G going to v Any edge in T which is not on the boundary δ M can intersect any edge in E G

14. Proof (III) (G, R) is weakly realizible if and only if (H, P) is weakly realizible. Considering the realization of H with the fewest intersections: By theorem 4.1, there is a realization such that there are at most intersections.

15. Theorem The weak realizability problem is in NP.

16. Proof Assume that (G, R) is weakly realizable. – By proposition 4.2, it has a weak realization with holes. – By lemma 4.3, there is a weak realization with holes in whiche each edge of triangulation T is intersected at most 2 12n+m times. – For any arc γ and edge, then the binary encoding of the number of intersections between γ and e is polynomial in n.

17. Proof (II) To verify weak realizibiltity with holes, guess for each edge of G: a labeling of the edges of T. By Lemma 3.5, for any it is possible to check in polynomial time that e and f are isotopically disjoint for the guessed set of lablings. By Lemma 3.2, this will guarantee a weak realization of (G, R)

18. Conclusions The string graph problem is NP-Complete The weak realizability problem is NP-Complete The pairwise crossing number problem, the existential theory of diagrams, and the existential fragment of topological inference are NP-Complete.