Vertex connectivity

Vertex connectivity $\kappa (G)$ is the smallest number of vertices whose deletion disconnects a nontrivial graph or reduces it to one vertex.

A graph with more than $k$ vertices is $k$-connected when deleting fewer than $k$ vertices always leaves it connected, equivalently when its vertex connectivity is at least $k$.

A $k$-connected graph has a cycle of length at least $\min \{|G|,2k\}$.

A component outside a longest cycle has at least $k$ attachment vertices in a $k$-connected graph, and no two attachments are consecutive, forcing cycle length at least $2k$.

A graph is 3-connected when it has at least four vertices and remains connected after deletion of any two vertices.