6The dicycle transversal number τ (D) of a digraph D is the minimum 7 size of a dicycle transversal of D, i. e. a set T ⊆ V (D) such that D − T 8 is acyclic. We study the following problem: Given a digraph D, decide 9 if there is a dicycle B in D and a cycle C in its underlying undirected 10 graph UG(D) such that V (B) ∩ V (C) = ∅. It is known that there is a 11 polynomial time algorithm for this problem when restricted to strongly 12 connected graphs, which actually finds B, C if they exist. We generalize 13 this to any class of digraphs D with either τ (D) = 1 or τ (D) = 1 and 14 a bounded number of dicycle transversals, and show that the problem is 15 N P-complete for a special class of digraphs D with τ (D) = 1 and, hence, 16 in general. 17 All graphs and digraphs are supposed to be finite, and they may contain loops 22 or multiple arcs or edges. Notation follows [1], and we recall the most relevant 23 concepts here. In order to distinguish between directed cycles in a digraph D 24 and cycles in its underlying graph UG(D) we use the name dicycle for a directed 25 cycle in D and cycle for a cycle in UG(D). Whenever we consider a (directed) 26 path P containing vertices a, b such that a precedes b on P , we denote by 27 P [a, b] the subpath of P which starts in a and ends in b. Similarly, we denote by 28 P (a, b], P [a, b), and P (a, b), respectively, the subpath that starts in the successor 29 of a on P and ends in b, starts in a and ends in the predecessor of b, and starts 30 in the successor of a on P and ends in the predecessor of b, respectively. The 31 same notation applies to dicycles. 32 arXiv:1106.5885v1 [math.CO]An in-tree (out-tree) rooted at a vertex r in a digraph D is a tree in UG(D) 1 whose arcs are oriented towards (away from) the root in D. 2A digraph D is acyclic if it does not contain a dicycle, and it is intercyclic if 3 it does not contain two disjoint dicycles. A dicycle transversal of D is a set S 4 of vertices of D such that D − S is acyclic, and the dicycle transversal number 5 τ (D) is defined to be the size of a smallest dicycle transversal. McCuaig 6 characterized the intercyclic digraphs of minimal in-and out-degree at least 2 7 in terms of their dicycle transversal number and designed a polynomial time 8 algorithm that, for any digraph, either finds two disjoint cycles or a structural 9 certificate for being intercyclic [7]. 10Theorem 1 [7] There exists a polynomial time algorithm which decides whether 11 a given digraph is intercyclic and finds two disjoint cycles if it is not. 12The undirected graphs without two disjoint cycles have been characterized by 13 Lovász [6], generalizing earlier statements of Dirac for the 3-connected case 14 [4]. The characterization again implies a polynomial algorithm for finding such 15 cycles if they exists.16Here we are concerned with the following problem. 17 Problem 1 Given a digraph D, decide if there is a dicycle B in D and a cycle 18 C in UG(D) with V (B) ∩ V (C) = ∅. 19 The motivation for studying this problem comes from [2] where ...

An out-(in-)branching B + s (B − s ) rooted at s in a digraph D is a connected spanning subdigraph of D in which every vertex x = s has precisely one arc entering (leaving) it and s has no arcs entering (leaving) it. We settle the complexity of the following two problems:• Given a 2-regular digraph D, decide if it contains an out-branching B + u such that D remains connected after removing the arcs of B + u .Both problems are NP-complete for general digraphs [1,5]. We prove that the first problem remains NPcomplete for 2-regular digraphs, whereas the second problem turns out to be polynomial when we do not prescribe the root in advance. We also prove that, for 2-regular digraphs, the latter problem is in fact equivalent to deciding if D contains two arc-disjoint out-branchings. We generalize this result to k-regular digraphs where we want to find a number of pairwise arc-disjoint spanning trees and out-branchings such that there are k in total, again without prescribing any roots.

A well known result in graph algorithms, due to Edmonds, states that given a digraph D and a positive integer , we can test whether D contains arc-disjoint out-branchings in polynomial time. However, if we ask whether there exists an outbranching and an in-branching which are arc-disjoint, then the problem becomes NP-complete. In fact, even deciding whether a digraph D contains an out-branching which is arc-disjoint from some spanning tree in the underlying undirected graph remains NP-complete. In this paper we formulate some natural optimization questions around these problems and initiate its study in the realm of parameterized complexity. More precisely, the problems we study are the following: Arc-Disjoint Branchings and Non-Disconnecting Out-Branching. In Arc-Disjoint Branchings (NonDisconnecting Out-Branching), a digraph D and a positive integer k are given as input and the goal is to test whether there exist an out-branching and in-branching (respectively, a spanning tree in the underlying undirected graph) that differ on at least k arcs. We obtain the following results for these problems.• Non-Disconnecting Out-Branching is fixed parameter tractable (FPT) and admits a linear vertex kernel. • Arc-Disjoint Branchings is FPT on strong digraphs. 123 AlgorithmicaThe algorithm for Non-Disconnecting Out-Branching runs in time 2 O(k) n O(1) and the approach we use to obtain this algorithms seems useful in designing other moderately exponential time algorithms for edge/arc partitioning problems.

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