Connectivity

Connectivity - связность.

The best known and most useful of the measures of how well a graph is conected is the connectivity, defined to be the minimum number of vertices in a set whose deletion results in a disconnected ortrivial graph.

Two vertices [math]\displaystyle{ u }[/math] and [math]\displaystyle{ v }[/math] in a graph [math]\displaystyle{ G }[/math] are said to be [math]\displaystyle{ k }[/math]-connected if there are [math]\displaystyle{ k }[/math] or more pairwise internally disjoint paths between them. The [math]\displaystyle{ (u,v) }[/math]-connectivity of [math]\displaystyle{ G }[/math], denoted [math]\displaystyle{ k_{G}(u,v) }[/math], is defined to be the maximum value of [math]\displaystyle{ k }[/math] for which [math]\displaystyle{ u }[/math] and [math]\displaystyle{ v }[/math] are [math]\displaystyle{ k }[/math]-connected.

If the order of [math]\displaystyle{ G }[/math] is [math]\displaystyle{ p }[/math], then the average connectivity of [math]\displaystyle{ G }[/math], denoted [math]\displaystyle{ \bar{k}(G) }[/math], is defined to be

[math]\displaystyle{ k_{G} = \frac{\sum_{u,v} k_{G}(u,v)}{\left(\begin{array}{c} p 2 \end{array}\right)}. }[/math]

(The expression [math]\displaystyle{ \sum_{u,v} k_{G}(u,v) }[/math] is sometimes referred to as the total connectivity of [math]\displaystyle{ G }[/math].) In contrast to the connectivity, which gives the smallest number of vertices whose failure disconnects some pair of vertieces, the average connectivity gives the expected number of vertices that must fail in order to disconnect an arbitrary pair of nonadjacent vertices.

See

• Edge connectivity, Vertex connectivity.