# Does a simple graph which consists of one vertex satisfy any edge or vertex connectivity?

I'm curious whether a simple graph which contains just one vertex is edge k-connected or vertex k-connected.

Edge-k-connectivity: We theoretically can remove any number of edges and it stays connected. The same with vertexes.

Graph satisfies is edge-k-connectivity: IF we remove any k edges -> Graph still satisfies connectivity.

In this case, the left side of implication is never satisfied so it is False. It the left side is False, then the whole implication is True.

EDIT: Graph -> simple graph

• Seems like a case of statements that are vacuously true. – A.Sh Jan 13 '16 at 9:08

• I'm not sure I agree: By convention the (vertex) connectivity of a complete graph $K_n$ is $n-1$. So, oddly enough, $K_1$ is an example of a connected graph with connectivity $0$. – Casteels Jan 13 '16 at 19:07