Towards Settling the Complexity of the Lettericity Problem

TL;DR

This paper explores the computational complexity of the lettericity problem, investigates retrieval problems, and introduces symmetric lettericity, linking it to neighborhood diversity and graph isomorphism.

Contribution

It proves polynomial-time solvability for word and decoder retrieval, shows coloring retrieval is as hard as graph isomorphism, and relates symmetric lettericity to neighborhood diversity.

Findings

Word and decoder retrieval are solvable in polynomial time.

Coloring retrieval is equivalent to the graph isomorphism problem.

Symmetric lettericity equals the neighborhood diversity of a graph.

Abstract

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.