arXiv2026
The lettericity of a graph $G$ is the smallest size of a set $Σ$ such that there exist $w_1, \ldots, w_{|V(G)|} \in Σ$ and a so-called decoder $D \subseteq Σ^2$ for which $G$ is isomorphic to the letter graph $(\{1, \ldots, |V(G)|\}, \{ij : 1 \le i < j \le |V(G)|, w_iw_j \in D\})$. It took around two decades of the study of lettericity for, in the simpler case of paths, a closed-form expression for its lettericity to be derived; this suggests that the question of whether the lettericity of an arbitrary graph can be computed in polynomial time is nontrivial. Indeed, this question has been raised repeatedly as an open problem in recent literature. We solve this problem by showing that the lettericity problem on arbitrary graphs is NP-complete. We also show that each of the natural ways in which a single part of a structure $(Σ, w_1, \ldots, w_{|V(G)|}, D)$ witnessing the lettericity could be constrained corresponds to an NP-complete problem, which resolves an additional three open problems from the literature; in particular, we show that the coloring extension, word extension, and decoder extension problems are NP-complete. Additionally, we use the reduction in the NP-completeness proof to give a lower bound on algorithms for lettericity under the Exponential Time Hypothesis (ETH): unless the ETH is false, there cannot exist a deterministic algorithm to decide whether the lettericity of an $n$-vertex graph is at most $k$ in time $2^{o(n)}$, even when $n = 6k$. Finally, we complement the known result that Lettericity is fixed-parameter tractable when parameterized by the standard parameter $k$ by providing a fixed-parameter algorithm with respect to the dual parameterization by $n-k$ running in $2^{\mathcal{O}((n-k)^2)} + \mathcal{O}(n)$ time.