Bidimensionality

1 PROBLEM DEFINITION The theory of bidimensionality provides general techniques for designing efficient fixed-parameter algorithms and approximation algorithms for a broad range of NP-hard graph problems in a broad range of graphs. This theory applies to graph problems that are " bidi-mensional " in the sense that (1) the solution value for the k ×k grid graph and similar graphs grows with k, typically as Ω(k 2), and (2) the solution value goes down when contracting edges and optionally when deleting edges in the graph. Many problems are bidimensional; a few classic examples are vertex cover, dominating set, and feedback vertex set. Graph classes. Results about bidimensional problems have been developed for increasingly general families of graphs, all generalizing planar graphs. The first two classes of graphs relate to embeddings on surfaces. A graph is planar if it can be drawn in the plane (or the sphere) without crossings. A graph has (Euler) genus at most g if it can be drawn in a surface of Euler characteristic g. A class of graphs has bounded genus if every graph in the class has genus at most g for a fixed g. The next three classes of graphs relate to excluding minors. Given an edge e = {v, w} in a graph G, the contraction of e in G is the result of identifying vertices v and w in G and removing all loops and duplicate edges. A graph H obtained by a sequence of such edge contractions starting from G is said to be a contraction of G. A graph H is a minor of G if H is a subgraph of some contraction of G. A graph class C is minor-closed if any minor of any graph in C is also a member of C. A minor-closed graph class C is H-minor-free if H / ∈ C. More generally, the term " H-minor-free " refers to any minor-closed graph class that excludes some fixed graph H. A single-crossing graph is a minor of a graph that can be drawn in the plane with at most one pair of edges crossing. A minor-closed graph class is single-crossing-minor-free if it excludes a fixed single-crossing graph. An apex graph is a graph in which the 1