Vertex cuts, edge cuts, vertex connectivity $\kappa(G)$ and edge connectivity $\lambda(G)$, with conventions for complete and one-vertex graphs

Let $G=(V,E)$ be connected with $n=|V|\ge2$. A vertex cut is a set $S\subseteq V$ for which $G-S$ is disconnected or has at most one vertex. An edge cut is a set $F\subseteq E$ for which $G-F$ is disconnected. The vertex connectivity and edge connectivity are

$\kappa(G):=\min\{\,|S|:S\text{ is a vertex cut}\,\},\qquad\lambda(G):=\min\{\,|F|:F\text{ is an edge cut}\,\}.$

The finite families being minimized are nonempty: deleting all but one vertex is a vertex cut, and deleting all edges is an edge cut. For the complete graph $K_n$, no deletion leaving at least two vertices disconnects it, so the first definition gives the standard exceptional convention $\kappa(K_n)=n-1$.

For the one-vertex graph $K_1$, set $\kappa(K_1)=\lambda(K_1)=0$. The null graph and every disconnected graph are outside the domain of these two connectivity parameters under this convention. Thus the inequalities involving $\delta(G)$ are stated only for nontrivial connected graphs, where the vertex set is nonempty and $\delta(G)$ is defined.