Refined Parameterizations for Computing Colored Cuts in Edge-Colored Graphs
Nils Morawietz, Niels Grüttemeier, Christian Komusiewicz, Frank O. Sommer · Theory of Computing Systems · 2022
Abstract In the NP-hard Colored (s,t)-Cut problem, the input is a graph G = (V,E) together with an edge-coloring ℓ : E → C, two vertices s and t, and a number k. The question is whether there is a set $S\subseteq C$ S ⊆ C of at most k colors such that deleting every edge with a color from S destroys all paths between s and t in G. We continue the study of the parameterized complexity of Colored (s,t)-Cut. First, we consider parameters related to the structure of G. For example, we study parameterization by the number ξi of edge deletions that are needed to transform G into a graph with maximum degree i. We show that Colored (s,t)-Cut is W[2]-hard when parameterized by ξ3, but fixed-parameter tractable when parameterized by ξ2. Second, we consider parameters related to the coloring ℓ. We show fixed-parameter tractability for three parameters that are potentially smaller than the total number of colors |C| and provide a linear-size problem kernel for a parameter related to the number of edges with rare edge colors.