families, has important algorithmic applications. Applications VLSI In VLSI design, the vertex separation problem was originally studied as a way to partition circuits into smaller subsystems, with a small number of components on the boundary between the subsystems. use interval thickness to model the number of tracks needed in a one-dimensional layout of a VLSI circuit, formed by a set of modules that need to be interconnected by a system of nets. In their model, one forms a graph in which the vertices represent nets, and in which two vertices are connected by an edge if their nets both connect to the same module; that is, if the modules and nets are interpreted as forming the nodes and hyperedges of a hypergraph then the graph formed from them is its line graph. An interval representation of a supergraph of this line graph, together with a coloring of the supergraph, describes an arrangement of the nets along a system of horizontal tracks (one track per color) in such a way that the modules can be placed along the tracks in a linear order and connect to the appropriate nets. The fact that interval graphs are perfect graphs implies that the number of colors needed, in an optimal arrangement of this type, is the same as the clique number of the interval completion of the net graph. Gate matrix layout is a specific style of CMOS VLSI layout for Boolean logic circuits. In gate matrix layouts, signals are propagated along "lines" (vertical line segments) while each gate of the circuit is formed by a sequence of device features that lie along a horizontal line segment. Thus, the horizontal line segment for each gate must cross the vertical segments for each of the lines that form inputs or outputs of the gate. As in the layouts of , a layout of this type that minimizes the number of vertical tracks on which the lines are to be arranged can be found by computing the pathwidth of a graph that has the lines as its vertices and pairs of lines sharing a gate as its edges. The same algorithmic approach can also be used to model folding problems in programmable logic arrays. Graph drawing Pathwidth has several applications to graph drawing: The minimal graphs that have a given crossing number have pathwidth that is bounded by a function of their crossing number. The number