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Cover (topology) In mathematics, and more particularly in set theory, a cover (or covering) of a set is a family of subsets of whose union is all of . More formally, if is an indexed family of subsets (indexed by the set ), then is a cover of if . Thus the collection is a cover of if each element of belongs to at least one of the subsets .
Definition. Formally, a vertex cover of an undirected graph is a subset of such that , that is to say it is a set of vertices where every edge has at least one endpoint in the vertex cover . Such a set is said to cover the edges of . The upper figure shows two examples of vertex covers, with some vertex cover marked in red.
In the matrix representation, an exact cover is a selection of rows such that each column contains a 1 in exactly one selected row. Each row represents a choice and each column represents a constraint. For example, the relation contains in the detailed example above can be represented by a 6×7 incidence matrix: 1. 2.
The set cover problem is a classical question in combinatorics, computer science, operations research, and complexity theory. Given a set of elements {1, 2, …, n} (called the universe) and a collection S of m subsets whose union equals the universe, the set cover problem is to identify the smallest sub-collection of S whose union equals the ...
A covering graph is a special case of a covering complex. [1] Both covering complexes and multigraphs with a 1-dimensional cell complex, are nothing but examples of covering spaces of topological spaces, so the terminology in the theory of covering spaces is available; say covering transformation group, universal covering, abelian covering, and ...
In topology, a covering or covering projection is a map between topological spaces that, intuitively, locally acts like a projection of multiple copies of a space onto itself. In particular, coverings are special types of local homeomorphisms. If is a covering, is said to be a covering space or cover of , and is said to be the base of the ...
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