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Concept
Permutation graph
Summary
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.
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.
A graph is a permutation graph if and only if both and its complement are comparability graphs.
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