By a constructive characterisation I mean a theorem giving a list of base graphs and a list of operations such that every graph in a given class is generated from the base graphs by applying some sequence of these operations and every intermediate graph is also in the class.
For 3-regular graphs I found the paper "Inductive classes of cubic graphs" by V Batagelj in 1981. http://vlado.fmf.uni-lj.si/vlado/papers/cubicEger.pdf
For 4-regular graphs there is a paper in French: "Construction of 4-regular graphs" by Bories and Jolivet in 1983.
I have been unable to find corresponding results for 5-regular graphs or 6-regular graphs or higher. It seems like a quite natural question, so I'd be surprised if nothing was known but I haven't been able to find any references myself.
Are there known constructive characterisations of k-regular (simple) graphs for any integer $k\geq 5$?
If not, can anyone give some intuition for why such results would be hard? Any interesting applications?