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Permutation graph

From Wikipedia, the free encyclopedia
The permutation graph and the matching diagram for the permutation (4,3,5,1,2)

In the mathematical field of graph theory, a permutation graph is a graph whose vertices represent the elements of a permutation, and whose edges represent pairs of elements that are reversed by the permutation. Permutation graphs may also be defined geometrically, as the intersection graphs of line segments whose endpoints lie on two parallel lines. Different permutations may give rise to the same permutation graph; a given graph has a unique representation (up to permutation symmetry) if it is prime with respect to the modular decomposition.[1]

Definition and characterization

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If is any permutation of the numbers from to , then one may define a permutation graph from in which there are vertices , and in which there is an edge for any two indices for which appears before in . That is, two indices and determine an edge in the permutation graph exactly when they determine an inversion in the permutation.

Given a permutation , one may also determine a set of line segments with endpoints and , such that . The endpoints of these segments lie on the two parallel lines and , and two segments have a non-empty intersection if and only if they correspond to an inversion in the permutation. Thus, the permutation graph of coincides with the intersection graph of the segments. For every two parallel lines, and every finite set of line segments with endpoints on both lines, the intersection graph of the segments is a permutation graph; in the case that the segment endpoints are all distinct, a permutation for which it is the permutation graph may be given by numbering the segments on one of the two lines in consecutive order, and reading off these numbers in the order that the segment endpoints appear on the other line.

Permutation graphs have several other equivalent characterizations:

  • A graph is a permutation graph if and only if is a circle graph that admits an equator, i.e., an additional chord that intersects every other chord.[2]
  • A graph is a permutation graph if and only if both and its complement are comparability graphs.[3]
  • A graph is a permutation graph if and only if it is the comparability graph of a partially ordered set that has order dimension at most two.[4]
  • If a graph is a permutation graph, so is its complement. A permutation that represents the complement of may be obtained by reversing the permutation representing .

Efficient algorithms

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It is possible to test whether a given graph is a permutation graph, and if so construct a permutation representing it, in linear time.[5]

As a subclass of the perfect graphs, many problems that are NP-complete for arbitrary graphs may be solved efficiently for permutation graphs. For instance:

Relation to other graph classes

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Permutation graphs are a special case of circle graphs, comparability graphs, the complements of comparability graphs, and trapezoid graphs.

The subclasses of the permutation graphs include the bipartite permutation graphs (characterized by Spinrad, Brandstädt & Stewart 1987) and the cographs.

Notes

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References

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  • Baker, Kirby A.; Fishburn, Peter C.; Roberts, Fred S. (1971), "Partial orders of dimension 2", Networks, 2 (1): 11–28, doi:10.1002/net.3230020103.
  • Bodlaender, Hans L.; Kloks, Ton; Kratsch, Dieter (1995), "Treewidth and pathwidth of permutation graphs", SIAM Journal on Discrete Mathematics, 8 (4): 606–616, doi:10.1137/S089548019223992X, hdl:1874/16657.
  • Brandstädt, Andreas; Le, Van Bang; Spinrad, Jeremy P. (1999), Graph Classes: A Survey, SIAM Monographs on Discrete Mathematics and Applications, ISBN 0-89871-432-X.
  • Dushnik, Ben; Miller, Edwin W. (1941), "Partially ordered sets", American Journal of Mathematics, 63 (3): 600–610, doi:10.2307/2371374, JSTOR 2371374.
  • Golumbic, Martin C. (1980), Algorithmic Graph Theory and Perfect Graphs, Computer Science and Applied Mathematics, Academic Press, p. 159.
  • McConnell, Ross M.; Spinrad, Jeremy P. (1999), "Modular decomposition and transitive orientation", Discrete Mathematics, 201 (1–3): 189–241, doi:10.1016/S0012-365X(98)00319-7, MR 1687819.
  • Spinrad, Jeremy P.; Brandstädt, Andreas; Stewart, Lorna K. (1987), "Bipartite permutation graphs", Discrete Applied Mathematics, 18 (3): 279–292, doi:10.1016/s0166-218x(87)80003-3.
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