Permutation graph

In mathematics, 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 σ = (σ₁,σ₂, ..., σ_n) is any permutation of the numbers from 1 to n, then one may define a permutation graph from σ in which there are n vertices v₁, v₂, ..., v_n, and in which there is an edge v_iv_j for any two indices i and j for which i < j and σ_i > σ_j. That is, two indices i and j determine an edge in the permutation graph exactly when they determine an inversion in the permutation.

Given a permutation σ, one may also determine a set of line segments s_i with endpoints (i,0) and (σ_i,1). The endpoints of these segments lie on the two parallel lines y = 0 and y = 1, 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 σ 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.

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