In the 1970s, Gy\H{o}ri and Lov\'{a}sz showed that for a $k$-connected $n$-vertex graph, a given set of terminal vertices $t_1, \dots, t_k$ and natural numbers $n_1, \dots, n_k$ satisfying $\sum_{i=1}^{k} n_i = n$, a connected vertex partition $S_1, \dots, S_k$ satisfying $t_i \in S_i$ and $|S_i| = n_i$ exists. However, polynomial algorithms to actually compute such partitions are known so far only for $k \leq 4$. This motivates us to take a new approach and constrain this problem to particular graph classes instead of restricting the values of $k$. More precisely, we consider $k$-connected chordal graphs and a broader class of graphs related to them. For the first, we give an algorithm with $O(n^2)$ running time that solves the problem exactly, and for the second, an algorithm with $O(n^4)$ running time that deviates on at most one vertex from the given required vertex partition sizes.

翻译：20世纪70年代，Győri和Lovász证明：对于$k$连通$n$顶点图，给定终端顶点集$t_1, \dots, t_k$和满足$\sum_{i=1}^{k} n_i = n$的自然数$n_1, \dots, n_k$，存在连通顶点划分$S_1, \dots, S_k$满足$t_i \in S_i$且$|S_i| = n_i$。然而，目前仅当$k \leq 4$时才存在多项式算法实际计算此类划分。这促使我们采用新方法，通过约束图类而非限制$k$值来研究该问题。具体而言，我们考虑$k$连通弦图及与之相关的更广泛图类。对于前者，我们给出时间复杂度为$O(n^2)$的精确求解算法；对于后者，我们给出时间复杂度为$O(n^4)$的算法，该算法在给定顶点划分规模上最多允许一个顶点的偏差。