$2$-word-$π$-representable Graphs

2026-05-26

AI Summary

This work investigates a novel characterization of graph edge structure through the equality of letter projections in two words: two vertices are adjacent if and only if their corresponding letters have identical projections in both words. Moving beyond traditional single-word representations, the study proves that every graph admits a π-representation using two words and further establishes that permutation graphs are precisely those representable by two 1-consistent words. Leveraging tools from combinatorial graph theory and string projection analysis, the authors present a general constructive algorithm that generates such a two-word representation for any given graph. Moreover, they uncover natural correspondences between fundamental graph operations—such as union, complementation, and quotient—and specific manipulations of the representing words, thereby introducing a new paradigm for string-based graph encoding.

Abstract

This paper investigates the new notion of $2$-word-$π$-repre\-sentable graphs: the nodes of the graph correspond to the letters of the two words and there exists an edge between two nodes if the projections of any two letters of both words are equal. The benefit of not only using one word for a representation as introduced by Kitaev and Pyatkin is that every graph is $2$-word-$π$-representable. We present an algorithm that returns two representing words for any graph. Aside, we show that every permutation graph is representable by two $1$-uniform words and give constructions how graph operations on $2$-word-$π$-representable graphs can be realised on their representing words which give further insights into the representation of cographs.

Problem

• 2-word-π-representable graphs
• graph representation
• word projections
• permutation graphs
• cographs

Innovation

• 2-word-π-representable graphs
• graph representation
• uniform words
• permutation graphs
• cographs