Towards Settling the Complexity of the Lettericity Problem

2026-05-08

AI Summary

The computational complexity of graph lettericity remains an open problem. This work investigates three variants of the lettericity reconstruction task—recovering one of the three components (word, decoder, or coloring) given a graph and any two of them—and introduces the notion of symmetric lettericity. By combining tools from combinatorial graph theory and graph isomorphism, the authors establish, for the first time, an equivalence between the coloring recovery problem and graph isomorphism. They further prove that symmetric lettericity coincides exactly with neighborhood diversity and is solvable in linear time. The main contributions include polynomial-time algorithms for both word and decoder recovery, as well as a linear-time algorithm for symmetric lettericity, thereby offering efficient new approaches to lettericity-related computations.

Abstract

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.

Problem

• lettericity
• graph complexity
• polynomial time
• graph isomorphism
• neighborhood diversity

Innovation

• lettericity
• symmetric lettericity
• neighborhood diversity
• graph isomorphism
• polynomial-time algorithm