TITLE: Are all almost regular graphs obvious?
QUESTION [11 upvotes]: Let the maximum and minimum degress of a graph be denoted (as usual) by $\Delta$ and $\delta$ respectively.

A graph is almost regular if $\Delta-\delta=1$.

Now, here is a simple way to generate such graphs: start with a regular graph and delete a matching. Either that or add a matching. 

An almost regular graph which is produced from a regular graph by the
  addition or removal of a matching is obvious.

Can you find examples of non-obvious almost regular graphs? So far, I am unable to produce any but I have a feeling there ought to be some.
A follow-up question, in case non-obvious ones do exist, would of course be to estimate which case is more prevalent.

REPLY [5 votes]: Here is a different generalization of $K_{2,3}$: Take any $3$-regular graph. Subdivide the edges into paths by adding an odd number of vertices. No vertices of degree $3$ in the resulting almost regular graph are adjacent to any other, so there is no way to remove a partial matching to produce a $2$-regular graph. The number of vertices of degree $2$ is odd so there is no matching on the set of vertices of degree $2$.
The same sort of construction can be used to make an almost regular graph $H$ from any regular graph $G$ of odd degree $d \gt 1$ so that $G$ is a minor of $H$ by stringing together $K_{d-1}$ graphs along each edge with an odd number of added vertices.