On Feedback Vertex Set: New Measure and New Structures

We present a new parameterized algorithm for 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 feedback vertex set of size $$k$$k that has no overlap with a given feedback vertex set $$F$$F of the graph $$G$$G. We develop an improved kernelization algorithm for disjoint-fvs and show that disjoint-fvs can be solved in polynomial time when all vertices in $$G{\setminus }F$$G\F have degrees upper bounded by three. We then propose a new branch-and-search process on disjoint-fvs, and introduce a new branch-and-search measure. The process effectively reduces a given graph to a graph on which disjoint-fvs becomes polynomial-time solvable, and the new measure more accurately evaluates the efficiency of the process. These algorithmic and combinatorial studies enable us to develop an $$O^*(3.83^k)$$O∗(3.83k)-time parameterized algorithm for the general fvs problem, improving all previous algorithms for the problem.