An improved FPT algorithm for Almost Forest Deletion problem

Abstract Almost Forest Deletion problem (AFD) is a generalization of the Feedback Vertex Set problem, which decides whether there exist at most k vertices in a given graph G whose removal from G results in an l -forest, where k and l are two given non-negative integers, and an l -forest is a graph which can be transformed into a forest by deleting at most l edges. Based on the iterative compression technique, we study the iterative version of the AFD problem, called Almost Forest Deletion Disjoint Compression problem (AFDDC), which asks for a new l -forest deletion set X ′ of size at most k for a given graph G that is disjoint with a given l -forest deletion set X of graph G for two given non-negative integers k and l . For the AFDDC problem, we develop some reduction rules to simplify a given instance, and give a new branching algorithm for the problem. A new branching measure is presented to evaluate the efficiency of the algorithm, which results in an algorithm of running time O ⁎ ( 4 k + l ) . Based on the proposed algorithm for the AFDDC problem, a parameterized algorithm for the AFD problem with running time O ⁎ ( 5 k 4 l ) is presented, improving the previous result O ⁎ ( 5.0024 ( k + l ) ) .