Connectedness - Connectivity

Properties and 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 create a disconnected graph. In recognition of this, such 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 create a disconnected graph. A 3-connected graph requires the removal of at least three vertices, and 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 notion of connectivity, the word may be used as a synonym for connectedness.

Another example of connectivity can be found in regular tilings. 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 equity is not kept)