From Few Components to an Eulerian Graph by Adding Arcs

Eulerian Extension (EE) is the problem to make an arcweighted directed multigraph Eulerian by adding arcs of minimum total cost. EE is NP-hard and has been shown fixed-parameter tractable with respect to the number of arc additions. Complementing this result, on the way to answering an open question, we show that EE is fixedparameter tractable with respect to the combined parameter “number of connected components in the underlying undirected multigraph” and “sum of indeg(v)− outdeg(v) over all vertices v in the input multigraph where this value is positive.” Moreover, we show that EE is unlikely to admit a polynomial-size problem kernel for this parameter combination and for the parameter “number of arc additions”.