String graph: Difference between revisions

Revision as of 23:52, 27 May 2025

Latest revision as of 11:59, 15 July 2025

(4 intermediate revisions by 3 users not shown)

Line 15:

If every edge of a given graph <math>G</math> is [[Subdivision (graph theory)|subdivided]], the resulting graph is a string graph if and only if <math>G</math> is planar. In particular, the subdivision of the [[complete graph]] <math>K_5</math> shown in the illustration is not a string graph, because <math>K_5</math> is not planar.<ref name="ss-s"/>

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|2010}} and {{harvtxt|Fox|Pach|2012}}.</ref>

Line 22:

{{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 highermore numbers of coloringscolors 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==

Line 57:

| title = Optimality program in segment and string graphs

| volume = 81

| year = 2019}}.| doi-access = free

}}.

*{{citation

| last1 = Chalopin | first1 = J. | last2 = Gonçalves | first2 = D. | last3 = Ochem | first3 = P.

Line 105 ⟶ 106:

| title = String graphs and incomparability graphs

| volume = 230

| year = 2012}}| hdl = 1721.1/98834

| hdl-access = free

}}.

*{{citation

| first1 = M. C. | last1 = Golumbic |author1-link = Martin Charles Golumbic | first2 = D. | last2 = Rotem | first3 = J. | last3 = Urrutia | author3-link = Jorge Urrutia Galicia

| title = Comparability graphs and intersection graphs

| journal = [[Discrete Mathematics (journal)|Discrete Mathematics]]