Towards Settling the Complexity of the Lettericity Problem

The lettericity of a graph $G=(V,E)$ is defined as the smallest size of an alphabet $\Sigma$ such that there is a word $w_1 \dots w_{|V|} \in \Sigma^*$ and a decoder $\mathcal{D} \subseteq \Sigma^2$ with the property that $G$ is isomorphic to the letter graph $G(\mathcal{D}, w)$, that is, the graph with vertex set $\{1, \dots, n\}$ and edge set $\{ij \mid 1\leq i < j \leq n, w_iw_j \in \mathcal{D}\}$. Note that $G(\mathcal{D}, w)$ can be seen as a graph with inherent coloring $\chi \colon V(G) \rightarrow \Sigma$. It is unknown whether the lettericity of a given graph can be computed in polynomial time. The problem to determine the lettericity of a given graph is called the lettericity problem. As a step towards answering the complexity of this problem, we investigate the following retrieval problems: given a graph $G$ together with two of the three solution-objects (word $w$, decoder $\mathcal{D}$, and coloring $\chi$), the goal is to compute the third solution-object. We show that word retrieval and decoder retrieval are solvable in polynomial time, while coloring retrieval is equivalent to the graph isomorphism problem. Beyond this, we introduce symmetric lettericity which is a restricted version of lettericity where each decoder needs to be symmetrical ($ab\in \mathcal{D}$ if and only if $ba\in \mathcal{D}$). As we show, the symmetric lettericity of a graph always equals the neighborhood diversity of the graph, which in fact can be computed in linear time.