Our newly developing theory of bidimensional graph problems provides
general techniques for designing efficient fixed-parameter algorithms and
approximation algorithms for NP-hard graph problems in broad classes of
graphs. This theory applies to graph problems that are
bidimensional in the sense that (1) the solution value for the k*k
grid graph (and similar graphs) grows with k, typically as
Omega(k²), and (2) the solution value goes down when contracting edges
and optionally when deleting edges. Examples of such problems include
feedback vertex set, vertex cover, minimum maximal matching, face cover, a
series of vertex-removal parameters, dominating set, edge dominating set,
r-dominating set, connected dominating set, connected edge dominating
set, connected r-dominating set, and unweighted TSP tour (a walk in the
graph visiting all vertices). Bidimensional problems have many structural
properties; for example, any graph embeddable in a surface of bounded
genus has treewidth bounded above by the square root of the problem's
solution value. These properties lead to efficient---often
subexponential---fixed-parameter algorithms, as well as polynomial-time
approximation schemes, for many minor-closed graph classes. One type of
minor-closed graph class of particular relevance has bounded local
treewidth, in the sense that the treewidth of a graph is bounded above in
terms of the diameter; indeed, we show that such a bound is always at most
linear. The bidimensionality theory unifies and improves several previous
results. The theory is based on algorithmic and combinatorial extensions
to parts of the Robertson-Seymour Graph Minor Theory, in particular
initiating a parallel theory of graph contractions. The foundation of this
work is the topological theory of drawings of graphs on surfaces.

This is from several joint papers mainly with Erik D. Demaine