# The Wikipedia definition of the degeneracy of a graph.

I'm trying to understand Wikipedia's definition of "Degeneracy (graph theory)" which says:

In graph theory, a k-degenerate graph is an undirected graph in which every subgraph has a vertex of degree at most k: that is, some vertex in the subgraph touches k or fewer of the subgraph's edges.

Degeneracy - Wikipedia

But then I look at the example and it does not make sense to me. Here is the graphic:

Now, from what I understand, a subgraph can be any subset of vertices and edges from the original graph. If so, one vertex can be a subgraph. If so, the 2-degenerate graph has many single vertices (aka subgraphs) that touch more than 2 edges.

I am sure I am missing something, but I don't know what it is.

• Also wiki definition only says each subgaph has a (i.e. at least one) vertex of degree $d \le k$ to be called $k$-degenerate. It doesn't matter if the subgraph has some other vertices of degree more than $k,$ only that at least one vertex in the subgraph has degree $d \le k.$ – coffeemath Nov 21 '18 at 18:38