graph theory In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...

, a connected

graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...

is said to be -vertex-connected (or -connected) if it has more than vertices and remains connected whenever fewer than vertices are removed. The vertex-connectivity, or just connectivity, of a graph is the largest for which the graph is -vertex-connected.

Definitions

A graph (other than a

complete graph In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices ...

) has connectivity ''k'' if ''k'' is the size of the smallest subset of vertices such that the graph becomes disconnected if you delete them. Complete graphs are not included in this version of the definition since they cannot be disconnected by deleting vertices. The complete graph with ''n'' vertices has connectivity ''n'' − 1, as implied by the first definition. An equivalent definition is that a graph with at least two vertices is ''k''-connected if, for every pair of its vertices, it is possible to find ''k'' vertex-independent paths connecting these vertices; see Menger's theorem . This definition produces the same answer, ''n'' − 1, for the connectivity of the complete graph ''K''_''n''. A 1-connected graph is called connected; a 2-connected graph is called biconnected. A 3-connected graph is called triconnected.

Applications

Polyhedral combinatorics

The 1- skeleton of any ''k''-dimensional convex polytope forms a ''k''-vertex-connected graph ( Balinski's theorem, ). As a partial converse, Steinitz's theorem states that any 3-vertex-connected

planar graph In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cro ...

forms the skeleton of a convex polyhedron.

Computational complexity

The vertex-connectivity of an input graph ''G'' can be computed in polynomial time in the following way''The algorithm design manual'', p 506, and ''Computational discrete mathematics: combinatorics and graph theory with Mathematica'', p. 290-291 consider all possible pairs

(s, t)

of nonadjacent nodes to disconnect, using Menger's theorem to justify that the minimal-size separator for

(s, t)

is the number of pairwise vertex-independent paths between them, encode the input by doubling each vertex as an edge to reduce to a computation of the number of pairwise edge-independent paths, and compute the maximum number of such paths by computing the maximum flow in the graph between

s

and

t

with capacity 1 to each edge, noting that a flow of

k

in this graph corresponds, by the integral flow theorem, to

k

pairwise edge-independent paths from

s

t

Notes

References