Refined Parameterizations for Computing Colored Cuts in Edge-Colored Graphs

Abstract

In the Colored \((s,t)\hbox {-}{\textsc {cut}}\) problem, the input is a graph \(G=(V,E)\) together with an edge-coloring \(\ell :E\rightarrow C\), two vertices s and t, and a number k. The question is whether there is a set \(S\subseteq C\) 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 \((s,t)\hbox {-}{\textsc {cut}}\). First, we consider parameters related to the structure of G. For example, we study parameterization by the number \(\xi _i\) of edge deletions that are needed to transform G into a graph with maximum degree i. We show that Colored \((s,t)\hbox {-}{\textsc {cut}}\) is \(\mathrm {W}[2]\)-hard when parameterized by \(\xi _3\), but fixed-parameter tractable when parameterized by \(\xi _2\). Second, we consider parameters related to the coloring \(\ell \). 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 a rare edge color.