On the problem of finding disjoint cycles and dicycles in a digraph

Abstract

We study the following problem: Given a digraph D, decide if there is a cycle B in D and a cycle C in its underlying undirected graph UG(D) such that V (B)∩V (C)=ø.

Whereas the problem is NP-complete if, as additional part of the input, a vertex x is prescribed to be contained in C, we prove that one can decide the existence of B,C in polynomial time under the (mild) additional assumption that D is strongly connected. Our methods actually find B,C in polynomial time if they exist. The behaviour of the problem as well as our solution depend on the cycle transversal number τ (D) of D, i.e. the smallest cardinality of a set T of vertices in D such that D-T is acyclic: If τ (D)≥3 then we employ McCuaig’s framework on intercyclic digraphs to (always) find these cycles. If τ (D) = 2 then we can characterize the digraphs for which the answer is “yes” by using topological methods relying on Thomassen’s theorem on 2-linkages in acyclic digraphs. For the case τ (D)≤1 we provide an algorithm independent from any earlier work.