Packing arc-disjoint cycles in oriented graphs

Abstract

Arc-Disjoint Cycle Packing is a classical NP-complete problem and we study it from two perspectives: (1) by restricting the cycles in the packing to be of a fixed length, and (2) by restricting the inputs to bipartite tournaments. Focusing first on Arc-Disjoint r-Cycle Packing (where the cycles in the packing are required to be of length r), we show NP-completeness in oriented graphs with girth r for each $r\ge 3$ and study the parameterized complexity of the problem with respect to two parameterizations (solution size and vertex cover size) for $r=4$ in oriented graphs. Moving on to Arc-Disjoint Cycle Packing in bipartite tournaments, we show that every bipartite tournament either contains k arc-disjoint cycles or has a feedback arc set of size at most $7(k-1)$. This result adds to the set of Erdös-Pósa-type results known in the combinatorics literature for packing and covering problems.