Packing directed circuits quarterintegrally The celebrated Erdxc5x91sPosa theorem states that every undirected graph that does not admit a family of k vertexdisjoint cycles contains a feedback vertex set (a set of vertices hitting all cycles in the graph) of size O(k log k). After being known for long as Youngeru0027s conjecture, a similar statement for directed graphs has been proven in 1996 by Reed, Robertson, Seymour, and Thomas. However, in their proof, the dependency of the size of the feedback vertex set on the size of vertexdisjoint cycle packing is not elementary.  :88,show that if we compare the size of a minimum feedback vertex set in a directed graph with emphquarterintegral cycle packing number, we obtain a polynomial bound. More precisely, we :88,show that if in a directed graph G there is no family of k cycles such that every vertex of G is in at most emphfour of the cycles, then there exists a feedback vertex set in G of size O(k4). On the way there we prove a more general result about quarterintegral packing of subgraphs of high directed treewidth: for every pair of positive integers a and b, if a directed graph G has directed treewidth Omega(a6 b8 log2(ab)), then one can find in G a family of a subgraphs, each of directed treewidth at least b, such that every vertex of G is in at most four subgraphs.

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.