Fixed-parameter single-exponential algorithm for lettericity

Determine whether the lettericity problem admits a deterministic algorithm running in time 2^{O(k)}n^{O(1)} for an n-vertex graph G and parameter k, given that the task is to decide whether the lettericity of G is at most k.

Background

The paper proves that deciding whether the lettericity of an n-vertex graph is at most k is NP-complete and, under the Exponential Time Hypothesis, cannot be solved in time 2^{o(n)}, even when n=6k. It also cites an existing algorithm with running time 2^{O(k^2 2^{2k})}n^3. The stated open problem asks whether this dependence on k can be improved to single-exponential form, 2^{O(k)}n^{O(1)}. Such an improvement would yield a 2^{O(n)} algorithm because instances with k≥n are trivially positive, thereby eliminating the logarithmic factor in the naive 2^{O(n log n)} approach and matching the ETH lower bound up to constants in the exponent.