# Parikh Word Representability of Bipartite Permutation Graphs

## TL;DR

This paper establishes that Parikh word representable graphs are exactly bipartite permutation graphs and explores their properties based on the arity of the representing words.

## Contribution

It extends the theory of Parikh word representable graphs beyond binary alphabets and characterizes them as bipartite permutation graphs.

## Key findings

- Parikh word representable graphs are equivalent to bipartite permutation graphs.
- Graph properties relate to the arity of the representing word.
- Theoretical development beyond binary alphabet cases.

## Abstract

The class of Parikh word representable graphs were recently introduced. In this work, we further develop its general theory beyond the binary alphabet. Our main result shows that this class is equivalent to the class of bipartite permutation graphs. Furthermore, we study certain graph theoretic properties of these graphs in terms of the arity of the representing word.