Thin graph classes and polynomial-time approximation schemes

Abstract

Baker [1] devised a powerful technique to obtain approx-imation schemes for various problems restricted to planar graphs. Her technique can be directly extended to vari-ous other graph classes, among the most general ones the graphs avoiding a fixed apex graph as a minor. Further gen-eralizations (e.g., to all proper minor closed graph classes) are known, but they use a combination of techniques and usually focus on somewhat restricted classes of problems. We present a new type of graph decompositions (thin sys-tems of overlays) generalizing Baker's technique and leading to straightforward polynomial-time approximation schemes. We also show that many graph classes (all proper minor-closed classes, and all subgraph-closed classes with bounded maximum degree and strongly sublinear separators) admit such decompositions.