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{{short description|Intersection graph for curves in the plane}}
In [[graph theory]], a '''string graph''' is an [[intersection graph]] of [[curve]]s 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''.▼
▲In [[graph theory]], a '''string graph''' is an [[intersection graph]] of [[Plane curve
== Background ==
{{harvs|txt|last=Benzer|first=Seymour|authorlink=Seymour Benzer|year=1959}} described a concept similar to string graphs as they applied to genetic structures. In that context, he also posed the specific case of intersecting intervals on a line, namely the now
through a collaboration between Sinden and [[Ronald Graham]], where the characterization of string graphs eventually came to be posed as an open question at the 5th Hungarian Colloquium on Combinatorics in 1976.<ref>{{harvtxt|Ehrlich|Even|Tarjan|1976}}, {{harvtxt|Graham|1976}}.</ref> However, the recognition of string graphs was eventually proven to be [[NP-complete]], implying that no simple characterization is likely to exist.
==Related graph classes==
[[File:Planar string graph.svg|thumb|300px|Representation of a [[planar graph]] as a string graph.]]
Every [[planar graph]] is a string graph:<ref name="ss-s">{{harvtxt|Schaefer|Štefankovič|2001}} credit this observation to {{harvtxt|Sinden|1966}}.</ref> one may form a string graph representation of an arbitrary plane-embedded graph by drawing a string for each vertex that loops around the vertex and around the midpoint of each adjacent edge, as shown in the figure. For any edge
[[Scheinerman's conjecture]], now proven, is the even stronger statement that every planar graph may be represented by the intersection graph of straight line segments, a very special case of strings.
[[File:Subdivided K5.svg|thumb|180px|A subdivision of ''K''<sub>5</sub> that is not a string graph.]]
If every edge of a given graph
Every [[circle graph]], as an intersection graph of line segments (the chords of a circle), is also a string graph. 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.{{sfnp|Kratochvíl|1991a|pp=56–57}}
The [[complement graph]] of every [[comparability graph]] is also a string graph.<ref>{{harvtxt|Golumbic|Rotem|Urrutia|1983}} and {{harvtxt|Lovász|1983}}. See also {{harvtxt|Fox|Pach|
==Computational complexity==
{{harvtxt|Kratochvíl|1991b}} showed string graph recognition to be [[NP-hard]], but was not able to show that it could be solved in [[NP (complexity)|NP]].{{sfnp|Kratochvíl|1991b}} One barrier to solving the problem in NP is that, for some string graphs, all systems of curves that realize the graph have an exponential number of crossings, so an explicit realization cannot be used as a polynomial-size witness for the graph being a string graph.{{sfnp|Kratochvíl|Matoušek|1991}} Instead, subsequent research in this area focused on compressed descriptions of realizations in terms of the sequences of crossings on each string, described using the theory of [[formal language]]s. After intermediate results by {{harvtxt|Schaefer|Štefankovič|2001}} and {{harvtxt|Pach|Tóth|2002}}, {{harvtxt|Schaefer|Sedgwick|Štefankovič|2003}} completed the proof that the problem is in NP, and therefore is [[NP-complete]].{{sfnp|Schaefer|Sedgwick|Štefankovič|2003}}
{{harvtxt|Ehrlich|Even|Tarjan|1976}} showed that testing whether a string graph is <math>k</math>-colorable is NP-complete, for every <math>k\ge 3</math>, and even when restricted to graphs with a given string representation consisting of straight line segments.{{sfnp|Ehrlich|Even|Tarjan|1976}} 3-colorings of string graphs, when they exist, can be found in the subexponential time bound <math>2^{O(n^{2/3}\log n)}</math>, but a similarly fast time for more colors is unlikely, under standard complexity-theoretic assumptions: an algorithm for 4-coloring in time <math>2^{o(n)}</math> would contradict the [[exponential time hypothesis]].{{sfnp|Bonnet|Rzążewski|2019}}
==Other results==
The smallest graph that is not a string graph has 12 vertices.{{sfnp|Kratochvíl|Goljan|Kučera|1986}}
{{harvtxt|Kratochvíl|1991a}} observed that [[Graph minor#Induced minors|induced minors]] of string graphs are also string graphs. Induced minors are obtained from a given graph by contracting edges and deleting vertices; unlike the more general form of graph minor they do not allow deleting edges. For graph minors, the [[Robertson–Seymour theorem]] states that any graph property closed under minors has finitely many [[Forbidden graph characterization|minimal forbidden minors]]. However, this does not hold for induced minors, and Kratochvíl found an infinite family of minimal forbidden induced minors for string graphs.{{sfnp|Kratochvíl|1991a}}
==Notes==
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[[Category:Topological graph theory]]
[[Category:Intersection classes of graphs]]
[[Category:NP-complete problems]]
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