Lettericity Is NP-Complete

Abstract

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 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 \textsf{NP}-complete (Theorem~10). We also prove that the coloring extension problem --- the same problem as lettericity, with the added condition that if $f$ is the isomorphism mapping from $G$ to the letter graph, $w_{f(v)} = χ(v)$ for a given coloring $χ$ of $G$ --- is \textsf{NP}-complete (Theorem~12). We also resolve the open problem of classifying the complexity of the word extension problem, which is the same problem as lettericity except that the $w_i$ are fixed; we show it to be \textsf{NP}-complete (Theorem~13), which, in tandem with our \textsf{NP}-completeness result for coloring extension, contrasts with the known result that when the constraint of the coloring extension problem and the constraint of the word extension problem are both applied to lettericity, lettericity can be decided in polynomial time. Additionally, we use the reduction in the \textsf{NP}-completeness proof to show that unless the Exponential Time Hypothesis 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 $n = 6k$ (Theorem~11).