arXiv MFCS 2024 | tags:[ graph theory ]

# Equitable Connected Partition and Structural Parameters Revisited: N-fold Beats Lenstra

with Dušan Knop, Jan Pokorný, and Šimon Schierreich## Abstract

We study the Equitable Connected Partition (ECP for short) problem, where we are given a graph $G=(V,E)$ together with an integer $p$, and our goal is to find a partition of $V$ into $p$ parts such that each part induces a connected sub-graph of $G$ and the size of each two parts differs by at most 1. On the one hand, the problem is known to be NP-hard in general and W[1]-hard with respect to the path-width, the feedback-vertex set, and the number of parts $p$ combined. On the other hand, fixed-parameter algorithms are known for parameters the vertex-integrity and the max leaf number.

As our main contribution, we resolve a long-standing open question [Enciso et al.; IWPEC ‘09] regarding the parameterisation by the tree-depth of the underlying graph. In particular, we show that ECP is W[1]-hard with respect to the 4-path vertex cover number, which is an even more restrictive structural parameter than the tree-depth. In addition to that, we show W[1]-hardness of the problem with respect to the feedback-edge set, the distance to disjoint paths, and NP-hardness with respect to the shrub-depth and the clique-width. On a positive note, we propose several novel fixed-parameter algorithms for various parameters that are bounded for dense graphs.