On the complexity of some cycle convexity parameters

The subject of graph convexity is well explored in the literature, the so-called interval convexities above all. In this work, we explore the cycle convexity, whose interval function is $I(S) = S \cup \{u \mid G[S \cup \{u\}]$ has a cycle containing $u\}$. In this convexity, we prove that the decision problems associated to the parameters rank and convexity number are in \NP-complete and \W[1]-hard when parameterized by the solution size. We also prove that to determine whether the percolation time of a graph is at least $k$ is \NP-complete, but polynomial for cacti or when $k\leq2$