Connected Partitions via Connected Dominating Sets

Abstract

The classical theorem due to Gy\H{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 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 Gy\H{o}ri--Lov\'{a}sz 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 Gy\H{o}ri--Lov\'{a}sz theorem: 1. On general graphs, a Gy\H{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.