关于几乎弦图的Gyubri-Lovász定理的有效构造
Efficient Constructions for the Győri-Lovász Theorem on Almost Chordal Graphs
摘要：
在20世纪70年代，Gyatri和Lovász证明了对于一个$k$ -连通的$n$ -顶点图，一个给定的满足$\sum_{i=1}^{k} n_i = n$的终端顶点$t_1, \dots, t_k$和自然数$n_1, \dots, n_k$的集合，存在一个满足$t_i \in S_i$和$|S_i| = n_i$的连通顶点划分$S_1, \dots, S_k$。然而，到目前为止，仅对于$k \leq 4$已知用于实际计算这样的分区的多项式算法。这促使我们采取一种新的方法，并限制这个问题，以特定的图形类，而不是限制$k$的值。更准确地说，我们认为$k$ -连接弦图和更广泛的一类图与他们有关。对于第一个，我们给出了一个算法与$O(n^2)$运行时间，解决了这个问题，第二，一个算法与$O(n^4)$运行时间，偏离最多一个顶点从给定的所需的顶点分区大小。
英文摘要：
In the 1970s, Győri and Lovász 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.
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