Carath\'eodory Number in Cycle Convexity

Let $G$ be a graph and $S\subseteq V(G)$. In the cycle convexity, we say that $S$ is $\mathit{\text{cycle}}\mathit{\text{convex}}$ if for any $u\in V(G)\setminus S$, the induced subgraph of $S\cup \{u\}$ contains no cycle that includes $u$. The $\mathit{\text{cycle}}\mathit{\text{convex}}\mathit{\text{hull}}$ of $S$, denoted by $\text{hullc}(S)$, is the smallest cycle convex set containing $S$. A set $S\subseteq V(G)$ is said to be $\mathit{\text{Carath}}\acute{\mathit{\text{e}}}\mathit{\text{odory}}\mathit{\text{independent}}$ if there exists a vertex $u\in \text{hullc}(S)$ such that $u\mathrm{\notin }{\displaystyle \bigcup _{a\in S}\text{hullc}(S\setminus \{a\})}$, and the Carath'eodory number $\text{car}(G)$ is the maximum size of such a set. In this paper, we prove that given a graph $G$ and $k\in \mathbb{N}$, deciding whether $\text{car}(G)\ge k$ is \NP-complete, even when $G$ is bipartite. On the other hand, we derive exact values and constant upper bounds for several graph classes, leading to polynomial-time algorithms. Some of them include forests, cycles, complete graphs, complete multipartite, split, and $P_4$-sparse graphs. In addition, we present a characterization of $n$-vertex graphs $G$ with extremal values near to $n$, including $\text{car}(G)=n-1$ and $\text{car}(G)=n-2$. Furthermore, we investigate the behavior of the Carath'eodory number under graph products such as the strong, lexicographic, and Cartesian products.