On Making a Distinguished Vertex Minimum Degree by Vertex Deletion

Abstract

For directed and undirected graphs, we study the problem to make a distinguished the unique minimum-(in) degree through deletion of a minimum of vertices. The corresponding NP-hard optimization problems are motivated by applications concerning control in elections and social network analysis. Continuing previous work for the directed case, we show that the problem is W[2]-hard when parameterized by the graph's arc set number, whereas it becomes fixed-parameter tractable when combining the parameters feed-back set and number of vertices to delete. For the so far unstudied undirected case, we show that the problem is NP-hard and W[1]-hard when parameterized by the number of vertices to delete. On the positive side, we show fixed-parameter tractability for several parameterizations measuring tree-likeness, including a vertex-linear problem kernel with respect to the parameter feedback edge set On the contrary, we show a non-existence result concerning polynomial-size problem kernels for the combined parameter vertex cover and of vertices to delete, implying corresponding nonexistence results when replacing cover by treewidth or set number.