Computing the Exchange Number in Graphs with respect to Cycle Convexity

Given a graph $G$, a subset $S\subseteq V(G)$ is $\mathit{\text{cycle}}\mathit{\text{convex}}$, if for any vertex $v\in V(G)\setminus S$, the induced subgraph, $G[S\cup \{v\}]$ cannot form a cycle containing the vertex $v$. The $\mathit{\text{exchange}}\mathit{\text{number}}$ of $G$, denoted by $e_{cc}(G)$ is the maximum cardinality of an \textit{E-independent} set of $G$. This paper studies the computational complexity of determining the exchange number of graphs and provides exact values for some graph classes. Given a graph $G$ and a positive integer $k$, we show that deciding whether $e_{cc}(G)\ge k$ is NP-complete even if $G$ is a $K_5$-free graph. In contrast, we characterize all $n$-vertex graphs $G$ with exchange number $n-1$ and obtain closed formulas for chordal graphs $G$ whose blocks lie in a single chain, which leads to polynomial-time algorithms for computing $e_{cc}(G)$. We also establish a lower bound for the exchange number of the Cartesian product of general graphs and by using the results of Anand et al. \cite{bijo2}, we derive an explicit formula for the exchange number of strong and lexicographic graph products.