The paper On graph classes with logarithmic boolean-width claims that some graph problems are fixed parameter tractable with parameter the boolean width.
In particular, boolean-width of the complement of $k$-degenerate graph is at most $k\log(n)$.
We got isomorphism preserving transformation graph $G$ to graph $G_m$ of logarithmic boolean width: set $G'_m$ to be the $m$-subdivision (i.e. subdivide each edge $m$ times) and set $G_m$ the complement of $G'_m$. $G'_m$ is $2$-degenerate.
$G_m$ is of logarithmic boolean width since its order is $O(m n^2)$ and its complement is $2$-degenerate.
For odd $m$, $G_m$ can be partitioned in two cliques (it is co-bipartite).
What is the intuition about so low boolean width for all transformed graphs?
Are there islands of tractable problems on $G_m$ related to problems on $G$?
Are there graph classes for which the transformation gives bounded boolean width?
