The paper [On graph classes with logarithmic
boolean-width](https://arxiv.org/abs/1009.0216) claims that the
boolean width of co-k-degenerate graphs is at most $k\log{n}$
and a lot of graph vertex partition problems can be solved in
polynomial time.

co-k-degenerate graphs include complements of bounded degree graphs.
Clique is NP-hard on co-maximum degree 4.

On the other hand,
[graphclasses.org](http://graphclasses.org/classes/par_21.html) claims
that clique is boolean width fixed parameter tractable, giving clique
width as reference. Since $\exp{\log{n}}=n$ it could be polynomial.

Are there complexity consequence of logarithmic boolean width of
co-bounded degree graphs? Like ETH not holding for them?

The paper gives polynomial algorithm for Dominating Set.
It claims some vertex problems are polynomial and some are $O(n^{\log(n)})$, p.2 for graphs of logarithmic boolean-width.

**Added** we learnt that this is known and the attack
doesn't work.

For details see Martin Vatshelle paper:
http://www.ii.uib.no/~martinv/Papers/MartinThesis.pdf
p. 58.

Someone please answer the question, don't feel like
answering my bounty.