Bidimensionality theory characterizes a broad range of graph problems (bidimensional) that admit efficient approximate, fixed-parameter or kernel solutions in a broad range of graphs. These graph classes include planar graphs, map graphs, bounded-genus graphs and graphs excluding any fixed minor. In particular, bidimensionality theory builds on the graph minor theory of Robertson and Seymour by extending the mathematical results and building new algorithmic tools. The theory was introduced in the work of Demaine, Fomin, Hajiaghayi, and Thilikos, for which the authors received the Nerode Prize in 2015.

A parameterized problem ${\displaystyle \Pi }$ is a subset of ${\displaystyle \Gamma ^{*}\times \mathbb {N} }$ for some finite alphabet ${\displaystyle \Gamma }$. An instance of a parameterized problem consists of (x,k), where k is called the parameter.

A parameterized problem ${\displaystyle \Pi }$ is minor-bidimensional if

Examples of minor-bidimensional problems are the parameterized versions of vertex cover, feedback vertex set, and longest path.

Let ${\displaystyle \Gamma _{r}}$ be the graph obtained from the ${\displaystyle (r\times r)}$-grid by triangulating internal faces such that all internal vertices become of degree 6, and then one corner of degree two joined by edges with all vertices of the external face. A parameterized problem ${\displaystyle \Pi }$ is contraction-bidimensional if

Examples of contraction-bidimensional problems are dominating set, connected dominating set, max-leaf spanning tree, and edge dominating set.

All algorithmic applications of bidimensionality are based on the following combinatorial property: either the treewidth of a graph is small, or the graph contains a large grid as a minor or contraction. More precisely,

Halin's grid theorem is an analogous excluded grid theorem for infinite graphs.