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 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 Gy\H{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 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.

翻译：Gy\H{o}ri 和 Lov\'{a}sz 的经典定理指出：对于任意 $k$-连通图 $G$，只要目标子图的总大小等于 $G$ 的大小，则 $G$ 可被划分为 $k$ 个连通子图，其中每个子图具有指定的大小并包含一个指定的顶点。然而，该结果在高效构造方面是众所周知的难题，目前尚不清楚此类划分是否能在多项式时间内计算，即使对于 $k = 5$ 的情况也是如此。我们通过考虑对 $k$-连通性要求进行自然弱化，朝着 Gy\H{o}ri--Lov\'{a}sz 定理的高效构造性版本取得进展。具体而言，我们证明：如果 $G$ 包含 $k$ 个不相交的连通支配集，则可以在多项式时间内找到所需的连通划分。基于这一结果，我们给出了原始 Gy\H{o}ri--Lov\'{a}sz 定理的若干高效近似和精确构造性版本：1. 对于一般图，当输入图的连通度为 $\Omega(k \cdot \log^2 n)$ 时，可在多项式时间内计算具有 $k$ 个部分的 Gy\H{o}ri--Lov\'{a}sz 划分；2. 对于凸二分图，$4k$ 的连通度即已足够；3. 对于双凸图和区间图，$k$ 的连通度即已足够，这意味着我们的算法在这些图类上给出了该定理的“真正”构造性版本。