Vertex-disjoint directed and undirected cycles in general digraphs

Abstract

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 ...

Restricted cycle factors and arc-decompositions of digraphs

“…As an example, in [2,3] the problem of deciding for a digraph D the existence of a directed cycle C in D and a cycle C ′ in U G(D) which are vertex disjoint was studied. It was shown that this problem is polynomially decidable for the class of digraphs with a bounded number of cycle transversals of size 1 (vertices whose removal eliminates all directed cycles) and N P-complete if we allow arbitrarily many transversal vertices.…”

“…It follows from (B.2) and (B.3) that B 3 = B ′′ [W \ {w 1 , w 2 } ∪ V 2 \ X] and B 4 = B ′′ [V 1 \ W ∪ X \ {x 1 , x 2 }] are balanced bipartite graphs with parts of size n/2 − 1 where each vertex has degree at least n/2 − 2. Hence, B 3 has a perfect matchingM 3 and B 4 has a perfect matchingM 4 . ThenM 3 ∪M 4 ∪ {x 1 w 1 , x 2 w 2 } is a perfect matching in B ′′ , contradicting that Problem 1.2 has a negative answer.…”

Arc‐Disjoint Directed and Undirected Cycles in Digraphs

“…Since a digraph with at least two nontrivial strong components of size greater than one has two disjoint dicycles and the dicyle transversal number of any digraph D is the sum of the dicycle transversal numbers of its strong components, we may assume that D has precisely one nontrivial strong component that is a no‐instance of Problem and since every digraph D with is a yes‐instance of Problem and hence also of Problem we can assume that .…”

“…In we completely characterized the complexity of the following problem that can be seen as lying in‐between the two problems of two disjoint cycles in a graph and two disjoint dicycles in a digraph. Problem Given a digraph D , decide if there is a dicycle B in D and a cycle C in with .…”

“…Theorem There is a polynomial time algorithm for Problem restricted to digraphs with a constantly bounded number of transversal vertices. The problem is NP‐complete for the class of all digraphs with .…”