# Connectivity (graph theory)

In mathematics and computer science, **connectivity** is one of the basic concepts of graph theory: it asks for the minimum number of elements (nodes or edges) that need to be removed to separate the remaining nodes into two or more isolated subgraphs.^{[1]} It is closely related to the theory of network flow problems. The connectivity of a graph is an important measure of its resilience as a network.

In an undirected graph G, two *vertices* u and v are called **connected** if G contains a path from u to v. Otherwise, they are called **disconnected**. If the two vertices are additionally connected by a path of length 1, i.e. by a single edge, the vertices are called **adjacent**.

A graph is said to be **connected** if every pair of vertices in the graph is connected. This means that there is a path between every pair of vertices. An undirected graph that is not connected is called **disconnected**. An undirected graph *G* is therefore disconnected if there exist two vertices in *G* such that no path in *G* has these vertices as endpoints. A graph with just one vertex is connected. An edgeless graph with two or more vertices is disconnected.

A directed graph is called **weakly connected** if replacing all of its directed edges with undirected edges produces a connected (undirected) graph. It is **unilaterally connected** or unilateral (also called **semiconnected**) if it contains a directed path from u to v or a directed path from v to u for every pair of vertices u, v.^{[2]} It is **strongly connected**, or simply strong, if it contains a directed path from u to v and a directed path from v to u for every pair of vertices u, v.

A **connected component** is a maximal connected subgraph of an undirected graph. Each vertex belongs to exactly one connected component, as does each edge. A graph is connected if and only if it has exactly one connected component.

The **strong components** are the maximal strongly connected subgraphs of a directed graph.

A **vertex cut** or **separating set** of a connected graph G is a set of vertices whose removal renders G disconnected. The **vertex connectivity** *κ*(*G*) (where G is not a complete graph) is the size of a minimal vertex cut. A graph is called ** k-vertex-connected** or

**if its vertex connectivity is k or greater.**

*k*-connectedMore precisely, any graph G (complete or not) is said to be *k-vertex-connected* if it contains at least *k*+1 vertices, but does not contain a set of *k* − 1 vertices whose removal disconnects the graph; and *κ*(*G*) is defined as the largest k such that G is k-connected. In particular, a complete graph with n vertices, denoted K_{n}, has no vertex cuts at all, but *κ*(*K _{n}*) =

*n*− 1.

A **vertex cut** for two vertices u and v is a set of vertices whose removal from the graph disconnects u and v. The **local connectivity** *κ*(*u*, *v*) is the size of a smallest vertex cut separating u and v. Local connectivity is symmetric for undirected graphs; that is, *κ*(*u*, *v*) = *κ*(*v*, *u*). Moreover, except for complete graphs, *κ*(*G*) equals the minimum of *κ*(*u*, *v*) over all nonadjacent pairs of vertices u, v.

2-connectivity is also called *biconnectivity* and 3-connectivity is also called *triconnectivity*. A graph G which is connected but not 2-connected is sometimes called *separable*.

Analogous concepts can be defined for edges. In the simple case in which cutting a single, specific edge would disconnect the graph, that edge is called a *bridge*. More generally, an edge cut of G is a set of edges whose removal renders the graph disconnected. The *edge-connectivity* *λ*(*G*) is the size of a smallest edge cut, and the local edge-connectivity *λ*(*u*, *v*) of two vertices u, v is the size of a smallest edge cut disconnecting u from v. Again, local edge-connectivity is symmetric. A graph is called *k-edge-connected* if its edge connectivity is k or greater.

A graph is said to be *maximally connected* if its connectivity equals its minimum degree. A graph is said to be *maximally edge-connected* if its edge-connectivity equals its minimum degree.^{[3]}

A graph is said to be *super-connected* or *super-κ* if every minimum vertex cut isolates a vertex. A graph is said to be *hyper-connected* or *hyper-κ* if the deletion of each minimum vertex cut creates exactly two components, one of which is an isolated vertex. A graph is *semi-hyper-connected* or *semi-hyper-κ* if any minimum vertex cut separates the graph into exactly two components.^{[4]}

More precisely: a G connected graph is said to be *super-connected* or *super-κ* if all minimum vertex-cuts consist of the vertices adjacent with one (minimum-degree) vertex.
A G connected graph is said to be *super-edge-connected* or *super-λ* if all minimum edge-cuts consist of the edges incident on some (minimum-degree) vertex.^{[5]}

A cutset X of G is called a non-trivial cutset if X does not contain the neighborhood N(u) of any vertex u ∉ X. Then the *superconnectivity* κ1 of G is:

A non-trivial edge-cut and the *edge-superconnectivity* λ1(G) are defined analogously.^{[6]}

One of the most important facts about connectivity in graphs is Menger's theorem, which characterizes the connectivity and edge-connectivity of a graph in terms of the number of independent paths between vertices.

If u and v are vertices of a graph G, then a collection of paths between u and v is called independent if no two of them share a vertex (other than u and v themselves). Similarly, the collection is edge-independent if no two paths in it share an edge. The number of mutually independent paths between u and v is written as *κ′*(*u*, *v*), and the number of mutually edge-independent paths between u and v is written as *λ′*(*u*, *v*).

Menger's theorem asserts that for distinct vertices *u*,*v*, *λ*(*u*, *v*) equals *λ′*(*u*, *v*), and if *u* is also not adjacent to *v* then *κ*(*u*, *v*) equals *κ′*(*u*, *v*).^{[7]}^{[8]} This fact is actually a special case of the max-flow min-cut theorem.

The problem of determining whether two vertices in a graph are connected can be solved efficiently using a search algorithm, such as breadth-first search. More generally, it is easy to determine computationally whether a graph is connected (for example, by using a disjoint-set data structure), or to count the number of connected components. A simple algorithm might be written in pseudo-code as follows:

By Menger's theorem, for any two vertices u and v in a connected graph G, the numbers *κ*(*u*, *v*) and *λ*(*u*, *v*) can be determined efficiently using the max-flow min-cut algorithm. The connectivity and edge-connectivity of G can then be computed as the minimum values of *κ*(*u*, *v*) and *λ*(*u*, *v*), respectively.

In computational complexity theory, SL is the class of problems log-space reducible to the problem of determining whether two vertices in a graph are connected, which was proved to be equal to L by Omer Reingold in 2004.^{[9]} Hence, undirected graph connectivity may be solved in O(log *n*) space.

The problem of computing the probability that a Bernoulli random graph is connected is called network reliability and the problem of computing whether two given vertices are connected the ST-reliability problem. Both of these are #P-hard.^{[10]}

The number of distinct connected labeled graphs with *n* nodes is tabulated in the On-Line Encyclopedia of Integer Sequences as sequence A001187. The first few non-trivial terms are