Is a k-connected graph also k-1 connected?

Question

So I'm reading about k-connected graphs, and I'm a little confused about them. This is the main definition I've seen:

A graph is k connected if and only if for any distinct x, y vertices in the graph, there are k internally disjoint paths from x to y in the graph

This makes me think that a k-connected graph is also a k-1 connected graph, since if it's k-connected there must exist k internally disjoint paths (which implies there must exist k-1 internally disjoint paths which means it must also be k-1 connected). Is this true? Or am I missing something?

Answer 1

Yes, this is the typical terminology. If you want to refer to the unique maximal k so that a graph is k-connected, you call it the connectivity. So a k-connected graph is (k-1)-connected, but a graph with connectivity k can't be said to have connectivity k-1.