The Forcing 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 \([S]=V(G)\) . A set \(S\) is a geodetic set if \(H(S)=V(G)\) ; while \(S\) is a hull set if \([S]=V(G)\) . The minimum cardinality of a geodetic set of \(G\) is the geodetic number \(g(G)\) . A subset \(T\) of a minimum hull set \(S\) is called a forcing subset for \(S\) if \(S\) is the unique minimum hull set containing \(T\) . The forcing hull number \(f(S,h)\) of \(S\) is the minimum cardinality among the forcing subsets of \(S\) , and the forcing hull number \(f(G,h)\) of \(G\) is the minimum forcing hull number among all minimum hull sets of \(G\) . The forcing geodetic number \(f(S,g)\)  of a minimum geodetic set \(S\) in \(G\) and the forcing geodetic number \(f(G,g)\) of \(G\) are defined in a similar fashion. The forcing hull numbers of several classes of graphs are determined. It is shown that for integers \(a,b\) with \(0\leq a\leq b\) , there exists a connected graph \(G\) such that \(f(G,h)=a\) and \(h(G)=b\) . We investigate the realizability of integers \(a,b\geq0\) that are the forcing hull and forcing geodetic numbers, respectively, of some graph.