Abstract
We study the Equitable Connected Partition problem. We examine the problem from the parameterized complexity perspective with respect to various (aggregate) parameterizations involving such secondary measurements as: (1) the number of partition classes, (2) the treewidth, (3) the pathwidth, (4) the minimum size of a feedback vertex set, (5) the minimum size of a vertex cover, (6) and the maximum number of leaves in a spanning tree of the graph. In particular, we show that the problem is W[1]-hard with respect to the first four combined, while it is fixed-parameter tractable with respect to each of the last two alone. The hardness result holds even for planar graphs. Furthermore, we show that the closely related problem of Equitable Coloring (equitably partitioning the vertices into a specified number of independent sets) is FPT parameterized by the maximum number of leaves in a spanning tree of the graph.