Efficient Constructions for the Győri-Lovász Theorem on Almost Chordal Graphs

Abstract

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 time algorithms to actually compute such partitions are known so far only for $k \le 4$. 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 class, we give an algorithm with $\mathcal {O}(n^2)$ running time that solves the problem exactly, and for the second, an algorithm with $\mathcal {O}(n^4)$ running time that deviates on at most one vertex from the required vertex partition sizes.