M_{2} (and, therefore, have linear-time algorithms on bounded treewidth graphs by the celebrated Courcelle’s theorem), but cannot be formulated in MSO_{1} (which would have yielded linear-time algorithms on bounded clique-width graphs by a well-known theorem of Courcelle, Makowsky, and Rotics). Each of these problems can be solved in time ^{f(k)} on graphs of clique-width ^{O(1)} assuming W[1]≠FPT. However, this does not rule out non-trivial improvements to the exponent ^{O(k)}, and proved that these problems cannot be solved in time ^{o(k)} unless ETH fails. Thus, prior to this work, E^{Θ (k)} algorithmic upper and lower bounds.

In this article, we provide lower bounds for H^{o(k)} matches asymptotically the recent upper bound ^{O(k)} due to Bergougnoux, Kanté, and Kwon (2017).

As opposed to the asymptotically tight ^{Θ(k)} bounds for E^{O(2k)} and a lower bound of merely ^{o(√ [4]k)} (implicit from the W[1]-hardness proof). In this article, we close the gap for G^{2o(k)}. This shows that G