Vertex-disjoint directed and undirected cycles in general digraphs

TL;DR

This paper investigates the computational complexity of finding disjoint directed and undirected cycles in general digraphs, extending known polynomial algorithms to broader classes and proving NP-completeness in some cases.

Contribution

It generalizes existing polynomial-time algorithms for specific digraph classes and establishes NP-completeness results for the problem in more general settings.

Findings

Polynomial-time algorithm for certain classes of digraphs

NP-completeness for digraphs with t(D)=1

Extension of cycle disjointness problem to broader graph classes

Abstract

The dicycle transversal number t(D) of a digraph D is the minimum size of a dicycle transversal of D, i. e. a set T of vertices of D such that D-T is acyclic. We study the following problem: Given a digraph D, decide if there is a dicycle B in D and a cycle C in the underlying undirected graph of D such such that B,C are disjoint. It is known that there is a polynomial time algorithm for this problem when restricted to strongly connected graphs, which actually finds B,C if they exist. We generalize this to any class of digraphs D with either t(D) not equal to 1 or t(D)=1 and a bounded number of dicycle transversals, and show that the problem is NP-complete for a special class of digraphs D with t(D)=1 and, hence, in general.