The Parameterized Complexity of Graph Cyclability

TL;DR

This paper investigates the computational complexity of determining the cyclability of graphs, proving hardness results and providing an fixed-parameter tractable algorithm for planar graphs based on advanced graph-theoretic techniques.

Contribution

It establishes the parameterized hardness of the cyclability problem and introduces an FPT algorithm for planar graphs using novel graph-theoretic insights.

Findings

Cyclability problem is co-W[1]-hard when parameterized by k.

No polynomial kernel exists for planar graphs unless NP ⊆ co-NP/poly.

An FPT algorithm for planar graphs with runtime 2^{2^{O(k^2 log k)}}·n^2.

Abstract

The cyclability of a graph is the maximum integer $k$ for which every $k$ vertices lie on a cycle. The algorithmic version of the problem, given a graph $G$ and a non-negative integer $k,$ decide whether the cyclability of $G$ is at least $k,$ is {\sf NP}-hard. We study the parametrized complexity of this problem. We prove that this problem, parameterized by $k,$ is ${\sf co\mbox{-}W[1]}$-hard and that its does not admit a polynomial kernel on planar graphs, unless ${\sf NP}\subseteq{\sf co}\mbox{-}{\sf NP}/{\sf poly}$. On the positive side, we give an {\sf FPT} algorithm for planar graphs that runs in time $2^{2^{O(k^2\log k)}}\cdot n^2$. Our algorithm is based on a series of graph-theoretical results on cyclic linkages in planar graphs.