String graph

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In graph theory, a string graph is an intersection graph of curves in the plane; each curve is called a "string". Given a graph G, G is a string graph if and only if there exists a set of curves, or strings, drawn in the plane such that no three strings intersect at a single point and such that the graph having a vertex for each curve and an edge for each intersecting pair of curves is isomorphic to G.

[edit] Background

In 1959, Benzer (1959) described a concept similar to string graphs as they applied to genetic structures. Later, Sinden (1966) specified the same idea to electrical networks and printed circuits. Through a collaboration between Sinden and Graham, the characterization of string graphs eventually came to be posed as an open question at the 5th Hungarian Colloquium on Combinatorics in 1976 and Elrich, Even, and Tarjan (1976) showed computing the chromatic number to be NP-hard. Kratochvil (1991) looked at string graphs and found that they form an induced minor closed class, but not a minor closed class of graphs and that the problem of recognizing string graphs is NP-hard. Schaeffer and Stefankovic (2002) found this problem to be NP-complete.

[edit] Related graph classes

Every planar graph is a string graph; Scheinerman's conjecture states that every planar graph may be represented by the intersection graph of straight line segments, which are a very special case of strings, and Chalopin, Gonçalves & Ochem (2007) proved that every planar graph has a string representation in which each pair of strings has at most one crossing point. Every circle graph, as an intersection graph of line segments, is also a string graph. And every chordal graph may be represented as a string graph: chordal graphs are intersection graphs of subtrees of trees, and one may form a string representation of a chordal graph by forming a planar embedding of the corresponding tree and replacing each subtree by a string that traces around the subtree's edges.

[edit] References

  • S. Benzer (1959). "On the topology of the genetic fine structure". Proceedings of the National Academy of Sciences of the United States of America 45: 1607--1620. 
  • F. W. Sinden (1966). "Topology of thin film RC-circuits". Bell System Technical Journal: 1639--1662. 
  • Chalopin, J.; Gonçalves, D.; Ochem, P. (2007). "Planar graphs are in 1-STRING". Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms: 609–617, ACM and SIAM. 
  • R. L. Graham (1976). "Problem 1". Open Problems at 5th Hungarian Colloquium on Combinatorics. 
  • G. Elrich and S. Even and R.E. Tarjan (1976). "Intersection graphs of curves in the plane". Journal of Combinatorial Theory 21. 
  • Jan Kratochvil (1991). "String Graphs I: The number of critical non-string graphs is infinite". Journal of Combinatorial Theory B 51. 
  • Jan Kratochvil (May 1991). "String Graphs II: Recognizing String Graphs is NP-Hard". Journal of Combinatorial Theory B 52: 67--78. 
  • Marcus Schaefer and Daniel Stefankovic (2001). "Decidability of string graphs". Proceedings of the 33rd Annual ACM Symposium on the Theory of Computing (STOC 2001): 241--246. 
  • Janos Pach and Geza Toth (2002). "Recognizing String Graphs is Decidable". Discrete and Computational Geometry 28: 593--606. 
  • Marcus Schaefer and Eric Sedgwick and Daniel Stefankovic (2002). "Recognizing String Graphs in NP". Proceedings of the 33th Annual Symposium on the Theory of Computing (STOC 2002).