A Single Exponential-Time FPT Algorithm for Cactus Contraction

TL;DR

This paper presents a fixed-parameter tractable algorithm with single exponential time complexity for the cactus contraction problem, expanding the class of graph modification problems efficiently solvable.

Contribution

It introduces a novel fixed-parameter algorithm for the extsc{Cactus Contraction} problem, previously unresolved in the parameterized complexity framework.

Findings

Algorithm runs in 2^{O(k)} * |V(G)|^{O(1)} time

Extends fixed-parameter tractability to cactus graphs

Builds on prior work for trees and paths

Abstract

For a collection $\mathcal{F}$ of graphs, the $\mathcal{F}$-\textsc{Contraction} problem takes a graph $G$ and an integer $k$ as input and decides if $G$ can be modified to some graph in $\mathcal{F}$ using at most $k$ edge contractions. The $\mathcal{F}$-\textsc{Contraction} problem is \NP-Complete for several graph classes $\mathcal{F}$. Heggerners et al. [Algorithmica, 2014] initiated the study of $\mathcal{F}$-\textsc{Contraction} in the realm of parameterized complexity. They showed that it is \FPT\ if $\mathcal{F}$ is the set of all trees or the set of all paths. In this paper, we study $\mathcal{F}$-\textsc{Contraction} where $\mathcal{F}$ is the set of all cactus graphs and show that we can solve it in $2^{\calO(k)} \cdot |V(G)|^{\OO(1)}$ time.