Efficient Constructions for the Gy\H{o}ri-Lov\'{a}sz Theorem on Almost Chordal Graphs

TL;DR

This paper develops efficient algorithms for constructing specific vertex partitions in $k$-connected chordal and related graphs, extending known results and providing practical solutions for cases where polynomial algorithms were previously limited.

Contribution

The paper introduces polynomial-time algorithms for partitioning $k$-connected chordal graphs, addressing cases beyond $k \, \leq 4$, with exact and near-exact solutions.

Findings

An $O(n^2)$ algorithm for exact partitions in $k$-connected chordal graphs.

An $O(n^4)$ algorithm for near-exact partitions in broader graph classes.

Extension of partitioning algorithms to larger classes of graphs.

Abstract

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.