On the Hull Number of a Graph

For two vertices \(u\) and \(v\) of a connected graph \(G\), the set \(H(u, v)\) consists of all those vertices lying on a \(u-v\) geodesic in \(G\). Given a set \(S\) of vertices of \(G\), the union of all sets \(H(u,v)\) for \(u,v \in S\) is denoted by \(H(S)\). A convex set \(S\) satisfies \(H(S) = S\). The convex hull \([S]\) is the smallest convex set containing \(S\). The hull number \(h(G)\) is the minimum cardinality among the subsets \(S\) of \(V(G)\) with \([S] = V(G)\). When \(H(S) = V(G)\), we call \(S\) a geodetic set. The minimum cardinality of a geodetic set is the geodetic number \(g(G)\). It is shown that every two integers \(a\) and \(b\) with \(2 \leq a \leq b\) are realizable as the hull and geodetic numbers, respectively, of some graph. For every nontrivial connected graph \(G\), we find that \(h(G) = h(G \times K_2)\). A graph \(F\) is a minimum hull subgraph if there exists a graph \(G\) containing \(F\) as induced subgraph such that \(V(F)\) is a minimum hull set for \(G\). Minimum hull subgraphs are characterized.