Circle graph

The intersection graph of chords of a circle: each chord is a vertex, and two vertices are adjacent when their chords cross. Two chords with distinct endpoints cross exactly when their endpoints alternate around the circle.

The intersection graph of the segments in a permutation diagram. Two segments cross exactly when their endpoint orders disagree. A set of mutually noncrossing segments corresponds to a monotone subsequence of the endpoint permutation.

A diagram of straight segments joining two parallel lines, with distinct endpoints on each line. Ordering the endpoints along the two lines encodes a permutation; crossing segments define the associated permutation graph.