# Clique-Width for Hereditary Graph Classes

## TL;DR

This paper surveys the known results on the boundedness of clique-width in hereditary graph classes and explores the implications for algorithmic problems like Colouring and Graph Isomorphism.

## Contribution

It provides a comprehensive overview of clique-width in hereditary classes and discusses the connection with well-quasi-orderability and algorithmic consequences.

## Key findings

- Bounded clique-width leads to polynomial algorithms for certain problems.
- Unbounded clique-width characterizes complexity in hereditary classes.
- Connections between clique-width and well-quasi-orderability are discussed.

## Abstract

Clique-width is a well-studied graph parameter owing to its use in understanding algorithmic tractability: if the clique-width of a graph class ${\cal G}$ is bounded by a constant, a wide range of problems that are NP-complete in general can be shown to be polynomial-time solvable on ${\cal G}$. For this reason, the boundedness or unboundedness of clique-width has been investigated and determined for many graph classes. We survey these results for hereditary graph classes, which are the graph classes closed under taking induced subgraphs. We then discuss the algorithmic consequences of these results, in particular for the Colouring and Graph Isomorphism problems. We also explain a possible strong connection between results on boundedness of clique-width and on well-quasi-orderability by the induced subgraph relation for hereditary graph classes.