Connected Partitions via Connected Dominating Sets
Aikaterini Niklanovits, Kirill Simonov, Shaily Verma, Ziena Zeif
TL;DR
This work addresses the constructive version of the Győri–Lovász theorem by tying GL-partitions to CDS partitions. It proves that if a graph $G$ contains a CDS partition of size $k$, a GL-partition can be computed in polynomial time, enabling efficient constructions under strengthened connectivity assumptions. The authors establish three main results: (i) GL-partitions with $k$ parts are computable in polynomial time for general graphs when $G$ is $Ω(k \cdot \log^2 n)$-connected; (ii) $4k$-connected convex bipartite graphs admit a CDS partition of size $k$ and hence a GL-partition; (iii) $k$-connected biconvex and interval graphs admit a CDS partition of size $k$, yielding true constructive GL-partitions on these classes. The approach relies on a CDS-to-GL reduction via a subroutine that builds a partial GL-partition and reduces the problem on the residual graph, linking connectivity, domination, and partitioning in a unified framework. Overall, the results enhance the practical applicability of GL-partitions and clarify the role of CDS partitions in enabling efficient graph partition constructions for restricted graph families.
Abstract
The classical theorem due to Győri and Lovász 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 long 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 Győri--Lovász theorem by considering a natural strengthening 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 Győri--Lovász theorem: 1. On general graphs, a Győri--Lovász partition with $k$ parts can be computed in polynomial time when the input graph has connectivity $Ω(k \cdot \log^2 n)$; 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.