Connectivity (graph theory)

In mathematics and computer science, connectivity is one of the basic concepts of graph theory. It is closely related to the theory of network flow problems. The connectivity of a graph is an important measure of its robustness as a network.

Definitions of components, cuts and connectivity

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. A graph is called connected if every pair of distinct vertices in the graph is connected (directly or indirectly). A connected component is a maximal connected subgraph of G. Each vertex belongs to exactly one connected component, as does each edge.

A directed graph is called weakly connected if replacing all of its directed edges with undirected edges produces a connected (undirected) graph. It is strongly connected or 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. The strong components are the maximal strongly connected subgraphs.

A cut or vertex cut of a connected graph G is a set of vertices whose removal renders G disconnected. The connectivity or vertex connectivity κ(G) is the size of a smallest vertex cut. A graph is called k-connected or k-vertex-connected if its vertex connectivity is k or greater. A complete graph with n vertices has no cuts at all, but by convention its connectivity is 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, κ(G) equals the minimum of κ(u,v) over all pairs of vertices u,v.

2-connectivity is also called "biconnectivity" and 3-connectivity is also called "triconnectivity".

Analogous concepts can be defined for edges. Thus 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.

Menger's theorem

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 greatest number of independent paths between u and v is written as κ′(u,v), and the greatest number of edge-independent paths between u and v is written as λ′(u,v).

Menger's theorem asserts that κ(u,v) = κ′(u,v) and λ(u,v) = λ′(u,v) for every pair of vertices u and v. This fact is actually a special case of the max-flow min-cut theorem.

Computational aspects

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.

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. Hence, undirected graph connectivity may be solved in $O(\log n)$ space.

Examples

• The vertex- and edge-connectivities of a disconnected graph are both 0.
• 1-connectedness is synonymous with connectedness.
• 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 every pair of vertices is 1.

Bounds on Connectivity

• The vertex-connectivity of a graph is less than or equal to its edge-connectivity. That is, κ(G) ≤ λ(G). Both are less than or equal to the minimum degree of the graph, since deleting all neighbors of a vertex of minimum degree will disconnect that vertex from the rest of the graph.

• For a vertex-transitive graph of degree d, we have: 2(d+1)/3 ≤ κ(G) ≤ λ(G) = d.

• For a vertex-transitive graph of degree d ≤ 4, or for any (undirected) Cayley graph of degree d, or for any symmetric graph of degree d, both kinds of connectivity are equal: κ(G) = λ(G) = d.

Other Properties

• Connectedness is preserved by graph homomorphisms.

• If G is connected then its line graph L(G) is also connected.

• If a graph G is k-connected, then for every set of vertices U of cardinality k, there exists a cycle in G containing U. The converse is true when k = 2.

• A graph G is 2-edge-connected if and only if it has an orientation that is strongly connected.

• The 1-skeleton of any $k$-dimensional convex polytope forms a $k$-vertex-connected graph. As a partial converse, Steinitz showed that any 3-vertex-connected planar graph forms the skeleton of a convex polyhedron.

See also

• Algebraic connectivity
• Small-world networks, Six degrees of separation, Small world phenomenon
• Scale-free network