Given a graph with labels defined on edges and a source-sink pair (s, t), the Label s-t Cut 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., l max , the maximum length of any s-t path (only applies to Label s-t Cut), and f max , the maximum number of appearances of any

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... label in the graph (applies to the two problems). We show that l max = 2 and f max = 2 are two complexity thresholds for Label s-t Cut. Furthermore, we give (i) an O * (c k ) time parameterized algorithm for Label s-t Cut with l max bounded from above, where parameter k is the number of labels in a solution, and c is a constant with l max − 1 < c < l max , (ii) a combinatorial l max -approximation algorithm for Label s-t Cut, and (iii) a polynomial time exact algorithm for Global Label Cut with f max bounded from above.