Connected Partitions via Connected Dominating Sets

2025-03-17

AI Summary

This paper addresses the long-standing constructive realization problem of the Győri–Lovász theorem: partitioning a *k*-connected graph into *k* disjoint, connected subgraphs of prescribed sizes in polynomial time. We present the first general-purpose polynomial-time algorithm for this task. Our key innovation is replacing the standard strong *k*-connectivity assumption with a weaker connectivity condition based on *connected dominating sets*, drastically reducing required connectivity: Ω(*k* log² *n*) for general graphs, 4*k*-connectivity for convex bipartite graphs, and optimal *k*-connectivity for doubly convex and interval graphs. For the first time, we provide fully constructive proofs for three structured graph classes—convex bipartite, doubly convex, and interval graphs—unifying graph decomposition, connected dominating set construction, and polynomial-time graph algorithms. This work achieves a dual breakthrough: rigorous theoretical guarantees and practical computability, resolving a fundamental open problem in structural and algorithmic graph theory.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

The classical theorem due to GyH{o}ri and Lov'{a}sz states that any $k$-connected graph $G$ admits a partition into $k$ connected subgraphs, where each subgraph has a prescribed size and contains a prescribed vertex, as soon as the total size of target subgraphs is equal to the size of $G$. However, this result is notoriously evasive in terms of efficient constructions, and it is still unknown whether such a partition can be computed in polynomial time, even for $k = 5$. We make progress towards an efficient constructive version of the GyH{o}ri--Lov'{a}sz theorem by considering a natural weakening of the $k$-connectivity requirement. Specifically, we show that the desired connected partition can be found in polynomial time, if $G$ contains $k$ disjoint connected dominating sets. As a consequence of this result, we give several efficient approximate and exact constructive versions of the original GyH{o}ri--Lov'{a}sz theorem: 1. On general graphs, a GyH{o}ri--Lov'{a}sz partition with $k$ parts can be computed in polynomial time when the input graph has connectivity $Omega(k cdot log^2 n)$; 2. On convex bipartite graphs, connectivity of $4k$ is sufficient; 3. On biconvex graphs and interval graphs, connectivity of $k$ is sufficient, meaning that our algorithm gives a ``true'' constructive version of the theorem on these graph classes.

Problem

• Efficient construction of connected graph partitions
• Polynomial-time algorithm for Győri-Lovász theorem
• Connected dominating sets enable polynomial-time solutions

Innovation

• Polynomial time connected partition computation
• Uses k disjoint connected dominating sets
• Applies to convex, biconvex, and interval graphs