Structural properties of graphs and eficient algorithms: Problems Between Parameters

Structural Properties of Graphs and Eficient Algorithms: Problems Between Parameters Dušan Knop Parameterized complexity became over last two decades one of the most impor- tant subfield of computational complexity. Structural graph parameters (widths) play important role both in graph theory and (parameterized) algoritmh design. By studying some concrete problems we exhibit the connection between struc- tural graph parameters and parameterized tractability. We do this by examining tractability and hardness results for the Target Set Selection, Minimum Length Bounded Cut, and other problems. In the Minimum Length Bounded Cut problem we are given a graph, source, sink, and a positive integer L and the task is to remove edges from the graph such that the distance between the source and the sink exceeds L in the resulting graph. We show that an optimal solution to the Minimum Length Bounded Cut problem can be computed in time f(k)n, where f is a computable function and k denotes the tree-depth of the input graph. On the other hand we prove that (under assumption that FPT ̸= W[1]) no such algorithm can exist if the parameter k is the tree-width of the input graph. Currently only few such problems are known. The Target Set Selection problem exibits the same phenomenon for the vertex cover number and..