On the vertex ranking problem for trapezoid, circular-arc and other graphs

We present polynomial time algorithms to solve the VERTEX RANKING problem for graphs of various graph classes among them trapezoid graphs, permutation graphs and circular-arc graphs. We demonstrate our approach in detail for a generalization of interval and trapezoid graphs called d-trapezoid graphs and for circular-arc graphs. All our algorithms use an approach called dynamic programming on pieces. Among others it exploits the property that all minimal separators and all the so-called pieces of these graphs can be represented by scanlines of an intersection model. This enables the design of polynomial time algorithms to list both all minimal separators and all pieces of an input graph.