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, an interval 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 determine whether the convexity number of a graph $G$ is at least $k$ is \NP-complete and \W[1]-hard when parameterized by the size of the solution when $G$ is a thick spider, but polynomial when $G$ is an extended $P_4$-laden graph. We also prove that determining whether the percolation time of a graph is at least $k$ is \NP-complete even for fixed $k \geq 9$, but polynomial for cacti or for fixed $k\leq2$.

翻译：图凸性这一主题在文献中已被充分探索，尤其是所谓的区间凸性。本文研究了循环凸性（一种区间凸性），其区间函数为 $I(S) = S \cup \{u \mid G[S \cup \{u\}]$ 包含一个含有 $u$ 的循环 $\}$。在该凸性下，我们证明了：对于厚蜘蛛图 $G$，判定其凸性数是否至少为 $k$ 是 \NP-完全的，且当以解的大小为参数时是 \W[1]-难的，但对于扩展的 $P_4$-负载图，该问题可在多项式时间内求解。此外，我们还证明了判定图的渗流时间是否至少为 $k$ 是 \NP-完全的（即使对于固定的 $k \geq 9$），但对于仙人掌图或固定的 $k \leq 2$ 时，该问题可在多项式时间内求解。