Partitioning a graph into small pieces with applications to path transversal

Abstract

Given a graph $$G = (V, E)$$ and an integer $$k \in \mathbb {N}$$, we study $$k$$-Vertex Separator (resp. $$k$$-Edge Separator), where the goal is to remove the minimum number of vertices (resp. edges) such that each connected component in the resulting graph has less than k vertices. We also study $$k$$-Path Transversal, where the goal is to remove the minimum number of vertices such that there is no simple path of length k. Our main results are the following improved approximation algorithms.