Length-Bounded Cuts: Proper Interval Graphs and Structural Parameters
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In the presented paper we study the Length-Bounded Cut problem for special graph classes as well as from a parameterized-complexity viewpoint. Here, we are given a graph G, two vertices s and t, and positive integers β and λ. The task is to find a set of edges F of size at most β such that every s-t-path of length at most λ in G contains some edge in F. Bazgan et al. conjectured that Length-Bounded Cut admits a polynomial-time algorithm if the input graph G is a proper interval graph. We confirm this conjecture by showing a dynamic-programming based polynomial-time algorithm. We strengthen the W[1]-hardness result of Dvořák and Knop. Our reduction is shorter, seems simpler to describe, and the target of the reduction has stronger structural properties. Consequently, we give W[1]-hardness for the combined parameter pathwidth and maximum degree of the input graph. Finally, we prove that Length-Bounded Cut is W[1]-hard for the feedback vertex number. Both our hardness results complement known XP algorithms.
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