On minimum bisection and related cut problems in trees and tree-like graphs

Minimum bisection denotes the NP-hard problem to partition the vertex set of a graph into two sets of equal sizes while minimizing the width of the bisection, which is defined as the number of edges between these two sets. It is intuitively clear that graphs with a somewhat linear structure are easy to bisect, and therefore our aim is to relate the minimum bisection width of a bounded-degree graph G to a parameter that measures the similarity between G and a path. First, for trees, we use the diameter and show that the minimum bisection width of every tree T on n vertices satisfies MinBis(T) <= 8n Delta(T)/diam(T). Second, we generalize this to arbitrary graphs with a given tree decomposition (T, chi) and give an upper bound on the minimum bisection width that depends on how close (T, chi) is to a path decomposition. Moreover, we show that a bisection satisfying our general bound can be computed in time proportional to the encoding length of the tree decomposition when the latter is provided as input.