Permutation graph

Definition and characterization

If ${\displaystyle \rho =({\sigma }_1,{\sigma }_2,...,{\sigma }_n)}$ is any permutation of the numbers from ${\displaystyle 1}$ to ${\displaystyle n}$, then one may define a permutation graph from ${\displaystyle \sigma }$ in which there are ${\displaystyle n}$ vertices ${\displaystyle v_1,v_2,...,v_n}$, and in which there is an edge ${\displaystyle v_i{v}_j}$ for any two indices ${\displaystyle i<j}$ for which ${\displaystyle j}$ appears before ${\displaystyle i}$ in ${\displaystyle \rho }$. That is, two indices ${\displaystyle i}$ and ${\displaystyle j}$ determine an edge in the permutation graph exactly when they determine an inversion in the permutation.

Given a permutation ${\displaystyle \sigma }$, one may also determine a set of line segments ${\displaystyle s_i}$ with endpoints ${\displaystyle (i,0)}$ and ${\displaystyle (k,1)}$, such that ${\displaystyle {\sigma }_k=i}$. The endpoints of these segments lie on the two parallel lines ${\displaystyle y=0}$ and ${\displaystyle y=1}$, 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 ${\displaystyle \sigma }$ 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 ${\displaystyle G}$ is a permutation graph if and only if ${\displaystyle G}$ is a circle graph that admits an equator, i.e., an additional chord that intersects every other chord.
• A graph ${\displaystyle G}$ is a permutation graph if and only if both ${\displaystyle G}$ and its complement ${\displaystyle \overline{G}}$ are comparability graphs.
• A graph ${\displaystyle G}$ 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.
• If a graph ${\displaystyle G}$ is a permutation graph, so is its complement. A permutation that represents the complement of ${\displaystyle G}$ may be obtained by reversing the permutation representing ${\displaystyle G}$.