Cycle transversals in bounded degree graphs

In this work we investigate the algorithmic complexity of computing a minimum C-k-transversal, i.e., a subset of vertices that intersects all the chordless cycles with k vertices of the input graph, for a fixed k >= 3. For graphs of maximum degree at most three, we prove that this problem is polynomial-time solvable when k <= 4, and NP-hard otherwise. For graphs of maximum degree at most four, we prove that this problem is NP-hard for any fixed k >= 3. We also discuss polynomial-time approximation algorithms for computing C-3-transversals in graphs of maximum degree at most four, based on a new decomposition theorem for such graphs that leads to useful reduction rules.