A Note on Finding Dual Feedback Vertex Set

TL;DR

This paper introduces algorithms for efficiently finding and enumerating small dual feedback vertex sets in edge-bicolored graphs, which hit all red and blue cycles, with fixed-parameter tractable time complexities.

Contribution

It presents fixed-parameter algorithms for finding and enumerating minimal dual feedback vertex sets in edge-bicolored graphs.

Findings

Dual feedback vertex set of size at most k can be found in O^*(c_1^k) time.

All minimal dual feedback vertex sets of size at most k can be enumerated in O^*(c_2^{k^2 + k}) time.

Algorithms use compact representations for efficient enumeration.

Abstract

For an edge-bicolored graph $G$ where each edge is colored either red or blue, a vertex set $S$ is a dual feedback vertex set if $S$ hits all blue cycles and red cycles of $G$. In this paper, we show that a dual feedback vertex set of size at most $k$ can be found in time $O^*(c_1^k)$ and all minimal dual feedback vertex set of size at most $k$ can be enumerated in time $O^*(c_2^{k^2 + k})$ by compact representations for constants $c_1$ and $c_2$.