On Feedback Vertex Set New Measure and New Structures

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We study the parameterized complexity of the feedback vertex set problem (fvs) on undirected graphs. We approach the problem by considering a variation of it, the disjoint feedback vertex set problem (disjoint-fvs), which finds a disjoint feedback vertex set of size k when a feedback vertex set of a graph is given. We show that disjoint-fvs admits a small kernel, and can be solved in polynomial time when the graph has a special structure that is closely related to the maximum genus of the graph. We then propose a simple branch-and-search process on disjoint-fvs, and introduce a new branch-and-search measure. The branch-and-search process effectively reduces a given graph to a graph with the special structure, and the new measure more precisely evaluates the efficiency of the branch-and-search process. These algorithmic, combinatorial, and topological structural studies enable us to develop an O(3.83 k kn 2) time parameterized algorithm for the general fvs problem, improving the previous best algorithm of time O(5 k k n 2) for the problem.