Vertex-disjoint directed and undirected cycles in general digraphs

Details Vertex-disjoint directed and undirected cycles in general digraphs Vertex-disjoint directed and undirected cycles in general digraphs

2011-06-29

arxiv.org/abs/1106.5885v1

Mathematics

Combinatorics

IMADA-preprint-math, 20 pages

Scientific paper

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.

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