Equitable Partitions into Matchings and Coverings in Mixed Graphs

Matchings and coverings are central topics in graph theory. The close relationship between these two has been key to many fundamental algorithmic and polyhedral results. For mixed graphs, the notion of matching forest was proposed as a common generalization of matchings and branchings. In this paper, we propose the notion of mixed edge cover as a covering counterpart of matching forest, and extend the matching--covering framework to mixed graphs. While algorithmic and polyhedral results extend fairly easily, partition problems are considerably more difficult in the mixed case. We address the problems of partitioning a mixed graph into matching forests or mixed edge covers, so that all parts are equal with respect to some criterion, such as edge/arc numbers or total sizes. Moreover, we provide the best possible multicriteria equalization.