In mathematics, connectedness is used to refer to various properties meaning, in some sense, "all one piece". When a mathematical object has such the property, we say it is for connected; otherwise it is disconnected. When a disconnected thing can be split naturally into connected pieces, regarded and identified separately. piece is ordinarily called a component or connected component.
Properties as well as parameters based on the idea of connectedness often involve the word connectivity. For example, in graph theory, a connected graph is one from which we must remove at least one vertex to take a disconnected graph. In recognition of this, such(a) graphs are also said to be 1-connected. Similarly, a graph is 2-connected if we must remove at least two vertices from it, to pull in a disconnected graph. A 3-connected graph requires the removal of at least three vertices, together with so on. The connectivity of a graph is the minimum number of vertices that must be removed to disconnect it. Equivalently, the connectivity of a graph is the greatest integer k for which the graph is k-connected.
While terminology varies, noun forms of connectedness-related properties often include the term connectivity. Thus, when discussing simply connected topological spaces, it is far more common to speak of simple connectivity than simple connectedness. On the other hand, in fields without a formally defined impression of connectivity, the word may be used as a synonym for connectedness.
Another example of connectivity can be found intilings. Here, the connectivity describes the number of neighbors accessible from a single tile:
3-connectivity in a triangular tiling,
4-connectivity in a square tiling,
6-connectivity in a hexagonal tiling,
8-connectivity in a square tiling note that distance equality is non kept