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Decidability of String Graphs

Decidability of String Graphs. Marcus Schaefer Daniel Stefankovic. v. v. string graph = intersection graph of a set of curves in the plane. Examples:. K is a string graph for any n. n. Any planar graph is a string graph. Examples:. K is a string graph for any n. n.

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Decidability of String Graphs

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  1. Decidability of String Graphs Marcus Schaefer Daniel Stefankovic v v

  2. string graph = intersection graph of a set of curves in the plane

  3. Examples: K is a string graph for any n n Any planar graph is a string graph

  4. Examples: K is a string graph for any n n Any planar graph is a string graph

  5. Given a graph G, decide if it is a string graph? ?

  6. Given a graph G, decide if it is a string graph? Not a string graph!

  7. History of the Question: S.Benzer: On the topology of the genetic fine structure ‘1959 F.W.Sinden: Topology of thin RC circuits ‘1966 conductors of with some capacitance between them do intersect, other conductors do not intersect

  8. Related older problem: Diagrammatic reasoning Is it possible that some A is B, some B is C, but no A is C?

  9. Is it possible that some A is B, some B is C, but no A is C? A C B

  10. ...has old history J. Vives J. Sturm G. Leibniz J. Lambert L. Euler

  11. Can any true statement of type be represented using diagrams? * “Is it possible that some A is B, some B is C, but no A is C?” (in the plane, each class being represented by a region homeomorphic to a disc) *) i.e. for any two concepts we specify whether they must/can/can not intersect.

  12. Can any true statement of type be represented using diagrams? “Is it possible that some A is B, some B is C, but no A is C?” NO

  13. Can we for any true statement of type decidewhether it can be represented by diagrams? “Is it possible that some A is B, some B is C, but no A is C?” ?

  14. NEXP

  15. Some known results: Ehrlich, Even, Tarjan ’76: computing the chromatic number of a string graph is NP-complete ` Kratochvil ’91: recognizing string graphs is NP-hard induced minor closed, infinitely many non-isomorphic forbidden induced minors

  16. An interesting question: v ` Kratochvil, Matousek ’91: Can we give an upper bound on the number of intersections of the smallest realization?

  17. Weak realizability: given a graph G and a set of pairs of edges R – is there a drawing of G in which only edges in R may intersect? e.g. for R=0 planarity

  18. string graph weak realizability any edge from and from and KM ‘91

  19. ` v Kratochvil, Matousek’91: Can we give an upper bound on the number of intersections of the smallest weak realization? SURPRISE![KM’91] There are graphs whose smallest weak representation has exponentially many intersections! Conjecture[KM’91]: at most exponentially many intersections

  20. Theorem: A graph with m edges has weak realization with at most m2 intersections. m Deciding string graphs is in NEXP.

  21. Given a graph G and pairs of edges which are allowed to intersect (some set R). (e.g. K with 2 edges allowed to intersect) 5 If (G,R) can be realized in the plane, can we give an upper bound on the number of intersections in the smallest realization?

  22. Idea: if there are too many intersections on an edge we will be able to redraw the realization to reduce the number of intersections. color the edges

  23. m suppose there are >2 intersections on e e (nontrivial = with >0 intersections) Then there is a non-trivial segment of e where each color occurs even number of times (possibly 0).

  24. m suppose there are >2 intersections on e e vector of parities of the colors to the left (2 pigeonholes) m Then there is a non-trivial segment of e where each color occurs even number of times (possibly 0).

  25. look at the segment: (a circle) axis (a mirror)

  26. number the intersections with circle: (a circle) 3 8 5 1 2 6 4 7 2-3,6-7,...,4k-2 – 4k-1- connected outside 4-5,...,4k – 4k+1- also connected outside

  27. look at the connections 2-3,6-7,...: (a circle) 1 3 8 3 5 1 1 3 4 2 2 6 4 4 2 7 (for all colors, respecting allowed intersections) 2-3,6-7,...,4k-2-4k-1- connected outside

  28. clear the inside and bring them inside (a circle) 1 3 8 3 5 1 1 3 4 2 2 6 4 4 2 7 (for all colors, respecting allowed intersections) 2-3,6-7,...,4k-2-4k-1- connected outside

  29. clear the inside and bring them inside (a circle) 8 1 3 3 5 1 3 1 2 4 6 2 4 2 4 7 (4-5,...,4k – 4k+1- connected outside)

  30. (a circle) 1 8 3 3 5 1 3 1 e 4 2 2 6 4 2 7 4 use mirror – now everything is connected. What about e?

  31. use upper or lower half of the circle as e 1 8 5 1 1 e 4 4 4 Decreased the # of intersections! (thus in a realization with minimal number of intersections <m2 of them) m

  32. consequences to topological inference: can decide realizability for more complex formulas: disjoint meet covered inside overlap in NEXP

  33. Conclusion: NEXP EXP PSPACE ? PH STRING GRAPHS NP

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