What Makes Equitable Connected Partition Easy Rosa Enciso 1 , Micheal R. Fellows 2 , Jiong Guo 3 , Iyad Kanj 4 , Frances Rosamond 2 , and Ondˇrej Such´ y 5 1 School of Electrical Engineering and Computer Science University of Central Florida, Orlando, FL renciso@cs.ucf.edu 2 University of Newcastle, Newcastle, Australia {michael.fellows|frances.rosamond}@newcastle.edu.au 3 Institut f¨ ur Informatik, Friedrich-Schiller-Universit¨at Jena, Ernst-Abbe-Platz 2, D-07743 Jena, Germany jiong.guo@uni-jena.de 4 School of Computing, DePaul University 243 S. Wabash Ave, Chicago, IL 60604 ikanj@cs.depaul.edu 5 Department of Applied Mathematics and Institute for Theoretical Computer Science Charles University, Malostransk´e n´am. 25, 118 00 Praha, Czech Republic suchy@kam.mff.cuni.cz Abstract. We study the Equitable Connected Partition problem, which is the problem of partitioning a graph into a given number of par- titions, such that each partition induces a connected subgraph, and the partitions differ in size by at most one. We examine the problem from the parameterized complexity perspective with respect to the number of partitions, the treewidth, the pathwidth, the size of a minimum feedback vertex set, the size of a minimum vertex cover, and the maximum num- ber 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 parameters (even combined), whereas it becomes fixed-parameter tractable when param- eterized by the last two parameters. The hardness result remains true even for planar graphs. We also show that the problem is in XP when parameterized by the treewidth (and hence any other mentioned struc- tural parameter). Furthermore, we show that the closely related problem, Equitable Coloring, is FPT when parameterized by the maximum number of leaves in a spanning tree of the graph.  Work partially supported by the ERASMUS program and by the DFG, project NI 369/4 (PIAF) while visiting Friedrich-Schiller-Universit¨at Jena (October 2008– March 2009), by grant 201/05/H014 of the Czech Science Foundation and by grant 1M0021620808 of the Czech Ministry of Education.