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 $3$.
We show that an identifying code of minimum order in the complementary prism of a cycle of order n has order 7n/9 + Θ(1). Furthermore, we observe that the clique-width of the complementary prism of a graph of clique-width k is at most 4k, and discuss some algorithmic consequences.
Let G be a finite, simple, and undirected graph and let S ⊆ V (G). In the geodetic convexity, S is convex if all vertices belonging to any shortest path between two vertices of S lie in S. The convex hull H(S) of S is the smallest convex set containing S. The hull number h(G) is the minimum cardinality of a set S ⊆ V (G) such that H(S) = V (G). The complementary prism GG̅ of a graph G arises from the disjoint union of the graph G and G̅ by adding the edges of a perfect matching between the corresponding vertices of G and G̅. Previous works have determined h(GG̅) when both G and G̅ are connected and partially when G is disconnected. In this paper, we characterize convex sets in GG̅ and we present equalities and tight lower and upper bounds for h(GG̅). This fills a gap in the literature and allows us to show that h(GG̅) can be determined in polynomial time, for any graph G.
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