The lettericity of a graph $G=(V,E)$ is defined as the smallest size of an alphabet $Σ$ such that there is a word $w_1 \dots w_{|V|} \in Σ^*$ and a decoder $\mathcal{D} \subseteq Σ^2$ 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 $χ\colon V(G) \rightarrow Σ$. 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 $χ$), 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.

翻译：图的文字性（lettericity）定义为：存在最小大小的字母表Σ，使得存在单词w₁…w|V|∈Σ*和译码器D⊆Σ²，图G=(V,E)同构于文字图G(D,w)——即顶点集为{1,…,n}、边集为{ij∣1≤i<j≤n, w_iw_j∈D}的图。注意，G(D,w)可视为带有固有着色χ: V(G)→Σ的图。当前未知给定图的文字性是否可在多项式时间内计算。确定给定图的文字性问题称为文字性问题。作为解答该问题复杂性的初步步骤，我们研究了以下检索问题：给定图G及其三个解对象中的两个（单词w、译码器D和着色χ），目标是计算第三个解对象。我们证明，单词检索与译码器检索可在多项式时间内求解，而着色检索等价于图同构问题。此外，我们引入了对称文字性——它是文字性的限制版本，要求每个译码器具有对称性（当且仅当ba∈D时，ab∈D）。研究表明，图的对称文字性始终等于其邻域多样性，而后者可在线性时间内计算。