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 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, 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.