On the geodetic hull number for complementary prisms II
In the geodetic convexity, a set of vertices $S$ of a graph $G$ is \textit{convex} if all vertices belonging to any shortest path between two vertices of $S$ lie in $S$. The \textit{convex hull} $H(S)$ of $S$ is the smallest convex set containing $S$. If $H(S) = V(G)$, then $S$ is a \textit{hull set}. The cardinality $h(G)$ of a minimum hull set of $G$ is the \textit{hull number} of $G$. The \textit{complementary prism} $G\overline{G}$ of a graph $G$ arises from the disjoint union of the graph $G$ and $\overline{G}$ by adding the edges of a perfect matching between the corresponding vertices of $G$ and $\overline{G}$. A graph $G$ is \textit{autoconnected} if both $G$ and $\overline{G}$ are connected. Motivated by previous work, we study the hull number for complementary prisms of autoconnected graphs. When $G$ is a split graph, we present lower and upper bounds showing that the hull number is unlimited. In the other case, when $G$ is a non-split graph, it is limited by $3$.