Extensions of permutation and interval graphs. (English) Zbl 0652.05055
Combinatorics, graph theory, and computing, Proc. 18th Southeast. Conf., Boca Raton/Fl. 1987, Congr. Numerantium 58, 267-275 (1987).
[For the entire collection see Zbl 0638.00009.]
Permutation graphs may be represented by a permutation diagram consisting of two parallel lines where one line has points numbered from 1 to n (the order of the graph) and the other line has points numbered by a permutation of (1,...,n). Vertex i is adjacent to vertex j iff the line joining the two i points intersects the line joining the two j points. We extend this notion to PI and II graphs in the following way: For a PI graph each vertex is represented by a point on one line and an interval on the other. Two vertices are adjacent iff their corresponding triangles intersect. In an II graph each vertex is represented by an interval on each line and two vertices are adjacent iff their corresponding trapezoids intersect. This extends the notion of Interval graphs where all intervals are on one line and two vertices are adjacent iff their corresponding intervals intersect. In this paper we illustrate some properties of PI and II graphs with respect to various classes of perfect graphs.