Recognizing Strings in NP Marcus Schaefer, Eric Sedgwick, Daniel Š tefankovi č

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

2. Definitions: • Given a collection of curves Ci where i I, the intersection graph is defined as (I, {(i, j) : Ci and Cj 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 2m 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 for the 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 EG going to v • Any edge in T which is not on the boundary δM can intersect any edge in EG

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 212n+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.