Tight bounds on the Carathéodory and exchange numbers in $\triangle$-convexity

Abstract

The $\triangle$-convexity space on a finite, simple graph $G = (V,E)$ is the collection $\mathcal{C}$ of subsets $S \subseteq V(G)$ such that whenever $x \in V(G)$ forms a triangle with two vertices in $S$, we have $x \in S$. The members of $\mathcal{C}$ are called convex sets, and the convex hull of a set $S \subseteq V(G)$, denoted $\operatorname{Hull}(S)$, is the smallest member of $\mathcal{C}$ that contains $S$. The Carathéodory (resp., exchange) number, $c_{\triangle}(G)$ (resp., $e_{\triangle}(G)$), is the size of a largest Carathéodory (resp., exchange) independent subset of $V(G)$. It was shown by Anand et al. (JCMCC 126, 2025, 11--27) that $c_{\triangle}(G) \leq t(G)+1$ and $e_{\triangle}(G) \leq t(G) + 2$, where $t(G)$ is the number of triangles in $G$, and that these bounds are tight. They also computed $c_{\triangle}(G)$ and $e_{\triangle}(G)$ for a block graph $G$ in terms of the number and arrangement of non-$K_2$ blocks in $G$. In this paper, we point out a gap in the proof in Anand et al. of the first inequality, $c_{\triangle}(G) \leq t(G)+1$, which has consequences for the proof of the second inequality, $e_{\triangle}(G) \leq t(G) + 2$, as well. Moreover, the tightness results are inadvertently applied as characterizations of the extremal graphs, leading to incorrect computations of the Carathéodory and exchange numbers of block graphs in certain cases. We fix these gaps by giving a full proof of the first inequality via a different route from that in Anand et al. Together with the argument in Anand et al., this also completes the proof of the second inequality. Our proof also leads to a characterization of the extremal graphs for each bound, which we use to compute the Carathéodory and exchange numbers of block graphs and to identify the extremal block graphs. We also determine $e_{\triangle}(G)$ exactly for $k$-trees for every $k \geq 2$.