The label cut problem with respect to path length and label frequency

Given a graph with labels defined on edges and a source-sink pair (s,t), the Labels-tCut problem asks for a minimum number of labels such that the removal of edges with these labels disconnects s and t. Similarly, the Global Label Cut problem asks for a minimum number of labels to disconnect G itself. For these two problems, we identify two useful parameters, i.e., lmax, the maximum length of any s-t path (only applies to Labels-tCut), and fmax, the maximum number of appearances of any label in the graph (applies to the two problems). We show that lmax=2 and fmax=2 are two complexity thresholds for Labels-tCut. Furthermore, we give (i) an O⁎(ck) time parameterized algorithm for Labels-tCut with lmax bounded from above, where parameter k is the number of labels in a solution, and c is a constant with lmax−1