# Multistage Vertex Cover

## TL;DR

This paper introduces Multistage Vertex Cover, a temporal graph problem requiring small vertex covers across layers with limited differences, revealing its computational hardness and fixed-parameter tractability results.

## Contribution

It initiates the study of Multistage Vertex Cover, analyzing its complexity and identifying fixed-parameter tractability in certain cases.

## Key findings

- Multistage Vertex Cover is NP-hard even in restricted settings.
- Certain parameterizations allow fixed-parameter tractability.
- The problem differs from classic and other dynamic vertex cover variants.

## Abstract

Covering all edges of a graph by a small number of vertices, this is the NP-complete Vertex Cover problem. It is among the most fundamental graph-algorithmic problems. Following a recent trend in studying temporal graphs (a sequence of graphs, so-called layers, over the same vertex set but, over time, changing edge sets), we initiate the study of Multistage Vertex Cover. Herein, given a temporal graph, the goal is to find for each layer of the temporal graph a small vertex cover and to guarantee that two vertex cover sets of every two consecutive layers differ not too much (specified by a given parameter). We show that, different from classic Vertex Cover and some other dynamic or temporal variants of it, Multistage Vertex Cover is computationally hard even in fairly restricted settings. On the positive side, however, we also spot several fixed-parameter tractability results based on some of the most natural parameterizations.