Dominated

Dominated is the past participle of the verb dominate, and in general usage describes a state in which someone or something is controlled, influenced, or outperformed by another. In everyday speech it can refer to social or political power, situational control, or a dominant performance, as in “the team dominated the game” or “the region was dominated by a single industry.”

In mathematics and related fields, dominated has more specialized senses. In real analysis, a function f is said to be dominated by a function g if |f(x)| ≤ g(x) for all x in its domain. When g is integrable, this domination is used in the dominated convergence theorem to justify exchanging limit and integral. The term also appears in the broader idea of domination in order theory and analysis, referring to a comparison that bounds or governs another object.

In graph theory, a vertex v dominates a vertex u if u is v or adjacent to v. A dominating set is a subset of vertices such that every vertex in the graph is either in the subset or adjacent to a vertex in the subset; a vertex is said to be dominated by the set if it is in the set or adjacent to a vertex in the set. This concept underpins problems in network control and facility location.

In ecology and biology, dominant species are those with the greatest influence on community structure due to abundance or biomass. The term “dominated” is used to describe species or populations that are suppressed or shaped by a dominant competitor or predator.