# A Comprehensive Introduction to the Theory of Word-Representable Graphs

## TL;DR

This paper provides a thorough overview of the theory of word-representable graphs, highlighting their properties, significance, and recent advancements in the field.

## Contribution

It offers a comprehensive introduction to word-representable graphs, covering foundational concepts and recent developments in the area.

## Key findings

- Word-representable graphs generalize several important graph classes.
- Existence of a word representing a given graph is characterized.
- Recent theoretical advancements have expanded understanding of these graphs.

## Abstract

Letters $x$ and $y$ alternate in a word $w$ if after deleting in $w$ all letters but the copies of $x$ and $y$ we either obtain a word $xyxy\cdots$ (of even or odd length) or a word $yxyx\cdots$ (of even or odd length). A graph $G=(V,E)$ is word-representable if and only if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $w$ if and only if $xy\in E$. Word-representable graphs generalize several important classes of graphs such as circle graphs, $3$-colorable graphs and comparability graphs. This paper offers a comprehensive introduction to the theory of word-representable graphs including the most recent developments in the area.