https://www.dagstuhl.de/22481

### November 27 – December 2 , 2022, Dagstuhl Seminar 22481

# Vertex Partitioning in Graphs: From Structure to Algorithms

## Organizers

Maria Chudnovsky (Princeton University, US)

Neeldhara Misra (Indian Institute of Techology – Madras, IN)

Daniel Paulusma (Durham University, GB)

Oliver Schaudt (Bayer AG – Leverkusen, DE)

## For support, please contact

Susanne Bach-Bernhard for administrative matters

Andreas Dolzmann for scientific matters

## Motivation

Many important discrete optimization problems can be modelled as graph problems that ask if the set of vertices in a graph can be partitioned into a smallest number of sets, such that each set has the same property, or into some number of sets, such that each set has a specific property of their own. This leads to a rich framework of vertex partitioning problems, which include classical problems such as Graph Colouring, Graph Homomorphism, Vertex Cover, Feedback Vertex Set, and Odd Cycle Transversal, and variants and generalizations of these problems, such as List Colouring, Connected Vertex Cover, Independent Feedback Vertex Set, and Subset Odd Cycle Transversal.

Most of these vertex partitioning problems are NP-hard. However, this situation may change if we insist that the input graph belongs to some special graph class. This leads to two fundamental questions, which lie at the heart of our Dagstuhl Seminar: for which graph classes can an NP-hard vertex partitioning problem be solved in polynomial time, and for which graph classes does the problem remain NP-hard?

In our seminar, we aim to discover, in a systematic way, general properties of graph classes from which we can determine the tractability or hardness of vertex partitioning problems. For this purpose, we bring together researchers from Discrete Mathematics and Theoretical Computer Science. Topics of the seminar will include:

- Vertex Partitioning Problems
- Hereditary Graph Classes
- Width Parameters
- Graph Decompositions
- Minimal Separators
- Parameterized Complexity

We plan an appropriate number of survey talks, presentations of recent results, open problem sessions, and ample time for discussions and problem solving. As concrete outcomes we expect to develop new, general methodology for solving vertex partitioning problems. This will also increase our understanding of how the complexities of these problems are related to each other when the input is restricted to some special graph class.

**Motivation text license**

Creative Commons BY 4.0

Maria Chudnovsky, Neeldhara Misra, Daniel Paulusma, and Oliver Schaudt

## Related Dagstuhl Seminar

- 19271: "Graph Colouring: from Structure to Algorithms" (2019)

## Classification

- Computational Complexity
- Data Structures And Algorithms
- Discrete Mathematics

## Keywords

- Computational complexity
- Hereditary graph classes
- Parameterized algorithms
- Polynomial-time algorithms
- Vertex partitioning