# Path-contractions, edge deletions and connectivity preservation

## TL;DR

This paper investigates the parameterized complexity of graph modification problems that preserve various forms of connectivity, revealing fixed-parameter tractability for biconnectivity deletion but hardness for strong connectivity preservation.

## Contribution

It proves that preserving strong connectivity via vertex deletion or path contraction is W[1]-hard, while biconnectivity preservation is fixed-parameter tractable with a polynomial kernel for the unweighted case.

## Key findings

- Biconnectivity deletion is fixed-parameter tractable with a $2^{O(k\log k)} n^{O(1)}$ algorithm.
- Preserving strong connectivity through vertex deletion or path contraction is W[1]-hard.
- Unweighted biconnectivity deletion admits a randomized polynomial kernel.

## Abstract

We study several problems related to graph modification problems under connectivity constraints from the perspective of parameterized complexity: {\sc (Weighted) Biconnectivity Deletion}, where we are tasked with deleting~$k$ edges while preserving biconnectivity in an undirected graph, {\sc Vertex-deletion Preserving Strong Connectivity}, where we want to maintain strong connectivity of a digraph while deleting exactly~$k$ vertices, and {\sc Path-contraction Preserving Strong Connectivity}, in which the operation of path contraction on arcs is used instead. The parameterized tractability of this last problem was posed by Bang-Jensen and Yeo [DAM 2008] as an open question and we answer it here in the negative: both variants of preserving strong connectivity are $\sf W[1]$-hard. Preserving biconnectivity, on the other hand, turns out to be fixed parameter tractable and we provide a $2^{O(k\log k)} n^{O(1)}$-algorithm that solves {\sc Weighted Biconnectivity Deletion}. Further, we show that the unweighted case even admits a randomized polynomial kernel. All our results provide further interesting data points for the systematic study of connectivity-preservation constraints in the parameterized setting.