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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.
1-connectedness is equivalent to connectedness for graphs of at least two vertices. The complete graph on n vertices has edge-connectivity equal to n − 1. Every other simple graph on n vertices has strictly smaller edge-connectivity. In a tree, the local edge-connectivity between any two distinct vertices is 1.
Connectedness features prominently in the definition of total orders: a total (or linear) order is a partial order in which any two elements are comparable; that is, the order relation is connected. Similarly, a strict partial order that is connected is a strict total order.
A graph with connectivity 4. In graph theory, a connected graph G is said to be k-vertex-connected (or k-connected) if it has more than k vertices and remains connected whenever fewer than k vertices are removed. The vertex-connectivity, or just connectivity, of a graph is the largest k for which the graph is k-vertex-connected.
As a consequence, a notion of connectedness can be formulated independently of the topology on a space. To wit, there is a category of connective spaces consisting of sets with collections of connected subsets satisfying connectivity axioms; their morphisms are those functions which map connected sets to connected sets (Muscat & Buhagiar 2006 ...
Local connectedness is, by definition, a local property of topological spaces, i.e., a topological property P such that a space X possesses property P if and only if each point x in X admits a neighborhood base of sets that have property P. Accordingly, all the "metaproperties" held by a local property hold for local connectedness. In particular:
In applied mathematics, lambda-connectedness (or λ-connectedness) deals with partial connectivity for a discrete space. Assume that a function on a discrete space (usually a graph ) is given. A degree of connectivity (connectedness) will be defined to measure the connectedness of the space with respect to the function.
Connectivity (graph theory), a property of a graph. The property of being a connected space in topology. Homotopical connectivity, a property related to the dimensions of holes in a topological space, and to its homotopy groups. Homological connectivity, a property related to the homology groups of a topological space.