Assume graphs of degree at most three for this question.

$k$-iso-regular graphs are the graphs in which every $k$-tuple has a same number of common neighbors. The $k$-dimensional  Weisfeiler Lehman fails on  $k$-iso-regular regular graphs. $k$ is a constant here, definitely if $k = O(n)$ then $k$-dimensional  Weisfeiler Lehman will work correctly.

Suppose I bound the maximum degree of the input graph to three, then also there are graphs on which $k$-dimensional  Weisfeiler Lehman fails. So one possible way to deal this situation is **individualization** along with $k$-dimensional  Weisfeiler Lehman.

Small example of an iso-regular graph: 

[![enter image description here][1]][1]

**Question :** Is there any known claim on the size of individualization set for $k$-iso-regular graphs ( degree at max three )? Is constant size individualization set possible? **It should be possible because I am here talking about graphs of degree at most three and there is an already polynomial time algorithm for bounded degree graph by E.M Luks.**

I tried to search on google scholar, but did not get anything specific.


  [1]: https://i.sstatic.net/DTk04.jpg