Subset feedback vertex set on graphs of bounded independent set size Abstract   The ( Weighted )  Subset Feedback Vertex Set  problem is a generalization of the classical  Feedback Vertex Set  problem and asks for a vertex set of minimum (weight) size that intersects all cycles containing a vertex of a predescribed set of vertices. Although  Subset Feedback Vertex Set  and  Feedback Vertex Set  exhibit different computational complexity on split graphs, no similar characterization is known on other classes of graphs. Towards the understanding of the complexity difference between the two problems, it is natural to study the importance of structural graph parameters. Here we consider graphs of bounded independent set number for which it is known that  Weighted Feedback Vertex Set  can be solved in polynomial time. We provide a dichotomy result with respect to the size xcexb1 of a maximum independent set. In particular we show that  Weighted Subset Feedback Vertex Set  can be solved in polynomial time for graphs with    xcexb1  xe2x89xa4  3   , whereas we prove that the problem remains NPhard for graphs with    xcexb1  xe2x89xa5  4   . Moreover, we show that the (unweighted)  Subset Feedback Vertex Set  problem can be solved in polynomial time on graphs of bounded independent set number by giving an algorithm with running time      n    O  (  xcexb1  )     . To complement our results, we demonstrate how our ideas can be extended to other terminal set problems on graphs of bounded independent set size.  Node Multiway Cut  is a terminal set problem that asks for a vertex set of minimum size that intersects all paths connecting any two terminals. Based on our findings for  Subset Feedback Vertex Set , we settle the complexity of  Node Multiway Cut  as well as its variants where nodes are weighted andor the terminals are deletable, for every value of the given independent set number.

Dynamic Parameterized Problems We study the parameterized complexity of various graph theoretic problems in the dynamic framework where the input graph is being updated by a sequence of edge additions and deletions. Vertex subset problems on graphs typically deal with finding a subset of vertices having certain properties. In real world applications, the graph under consideration often changes over time and due to this dynamics, the solution at hand might lose the desired properties. The goal in the area of dynamic graph algorithms is to efficiently maintain a solution under these changes. Recomputing a new solution on the new graph is an expensive task especially when the number of modifications made to the graph is significantly smaller than the size of the graph. In the context of parameterized algorithms, two natural parameters are the size k of the symmetric difference of the edge sets of the two graphs (on n vertices) and the size r of the symmetric difference of the two solutions. We study the Dynamic                                                                           Pi                                                                         :239,396,558,Deletion problem which is the dynamic variant of the classical                                                                           Pi                                                                         :239,396,558,Deletion problem and show NPhardness, fixedparameter tractability and kernelization results. For specific cases of Dynamic                                                                           Pi                                                                         :239,396,558,Deletion such as Dynamic Vertex Cover and Dynamic Feedback Vertex Set, we describe improved algorithms and linear kernels. Specifically, we show that Dynamic Vertex Cover has a deterministic algorithm with                                                                           1.0822k nmathcal                                                                          :736,897,1062,1232,running time and Dynamic Feedback Vertex Set has a randomized algorithm with                                                                           1.6667k nmathcal                                                                          :736,897,1062,1232,running time. We also show that Dynamic Connected Feedback Vertex Set can be solved in                                                                           2mathcal O(k) nmathcal                                                                          time. For each of Dynamic Connected Vertex Cover, Dynamic Dominating Set and Dynamic Connected Dominating Set, we describe an algorithm with                                                                           2k nmathcal                                                                          :736,897,1062,1232,running time and show that this is the optimal :736,897,1062,1232,running time (up to polynomial factors) assuming the Set Cover Conjecture.