# Complexity consequence of logarithmic boolean width of co-bounded degree graphs?

The paper On graph classes with logarithmic boolean-width 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 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.