\(\triangle\)-Convexity Hull Explorer

In the \(\triangle\)-convexity on a graph \(G\), the hull of a vertex set \(S\) is computed iteratively. Starting from \(I^0_\triangle(S)=S\), each level \(I^k_\triangle(S)\) adds every vertex outside the current set that forms a triangle with two vertices already inside. The process stabilises at \(\operatorname{Hull}(S)=I^\tau_\triangle(S)\), where \(\tau\) is the iteration time of \(S\).

A set \(S\) is Carathéodory-independent if \(\operatorname{Hull}(S)\setminus\bigcup_{a\in S}\operatorname{Hull}(S\setminus\{a\})\neq\emptyset\). Once the full hull is displayed, use Check Carathéodory to step through each element \(a\in S\), watching how the union of sub-hulls gradually covers \(\operatorname{Hull}(S)\).