Connected (Dense) Partition for Tree-Like Graphs

2026-07-24

AI Summary

This study addresses the problem of connected dense partitioning in graphs, which seeks to maximize the density of each part under constraints on either the number of parts or their sizes. The authors propose a dynamic programming framework based on tree decompositions, yielding the first polynomial-time algorithm for this problem on thick forests and extending it to general graphs of bounded treewidth. Key contributions include an efficient method for constructing partitions that simultaneously satisfy connectivity and high-density requirements, a proof establishing the NP-hardness of the problem on split graphs, and an effective instantiation of the partition structure guaranteed by the Győri–Lovász theorem.

Abstract

We focus on two variants of graph partitioning problems, connected partition and dense partition. Formally, given a graph $G=(V,E)$ and a partition of its vertices $\mathcal P=\{P_1,\ldots, P_k\}$ we say that $\mathcal P$ is a connected partition of $G$ if each $P_i$ induces a connected graph in $G$. Many classical variants of this problem impose additional restrictions both on the number of parts as well as on the size of each part. Moreover, given a partition $\mathcal P=\{P_1,\ldots, P_k\}$ we define its density by $d(\mathcal P):=\sum_{i=1}^k |E(P_i)|/|V(P_i)|$. The problem Maximum Dense Graph Partition asks to construct a partition of maximum density. We study this problem both with and without fixed number of sets $k$. We prove the following results: 1. A polynomial time algorithm for Maximum Dense Graph Partition of thick forests, a subclass of chordal graphs, generalizing the previously known polynomial time algorithm on block graphs. 2. A generic dynamic programming algorithm to construct (if possible) a connected partition into $k$ sets of prescribed sizes on graphs with bounded treewidth. This yields algorithms for both variants of Dense Graph Partition and an efficient construction for the Győri-Lovász theorem. 3. The $\mathsf{NP}$-hardness of Maximum Dense Graph Partition to $k$ parts restricted to split graphs, indicating that thick trees are the boundary for the polynomial computability of this problem.

Problem

• connected partition
• dense partition
• graph partitioning
• maximum density
• tree-like graphs

Innovation

• dense partition
• connected partition
• bounded treewidth
• polynomial-time algorithm
• NP-hardness