Summary: Let \(G\) be a graph. The connectivity of \(G\), \(\kappa(G)\), is the maximum integer \(k\) such that there exists a \(k\)-container between any two different vertices. A \(k\)-container of \(G\) between \(u\) and \(v\), \(C_k(u,v)\), is a set of \(k\)-internally-disjoint paths between \(u\) and \(v\). A spanning container is a container that spans \(V(G)\). A graph \(G\) is \(k^*\)-connected if there exists a spanning \(k\)-container between any two different vertices. The spanning connectivity of \(G\), \(\kappa^*(G)\), is the maximum integer \(k\) such that \(G\) is \(w^*\)-connected for \(1\leq w\leq k\) if \(G\) is \(1^*\)-connected.
Let \(x\) be a vertex in \(G\) and let \(U=\{y_1,y_2,\dots,y_k\}\) be a subset of \(V(G)\) where \(x\) is not in \(U\). A spanning \(k-(x,U)\)-fan, \(F_k(x,U)\), is a set of internally-disjoint paths \(\{P_1,P_2,\dots,P_k\}\) such that \(P_i\) is a path connecting \(x\) to \(y_i\) for \(1\leq i\leq k\) and \(\bigcup^k_{i=1}V(P_i)=V(G)\). A graph \(G\) is \(k^*\)-fan-connected (or \(k^*_j\)-connected) if there exists a spanning \(F_k(x,U)\)-fan for every choice of \(x\) and \(U\) with \(|U|=k\) and \(x\neq U\). The spanning fan-connectivity of a graph \(G\), \(\kappa^*_f(G)\), is defined as the largest integer \(k\) such that \(G\) is \(w^*_f\)-connected for \(1\leq w\leq k\) if \(G\) is \(1^*_f\)-connected.
In this paper, some relationship between \(\kappa(G)\), \(\kappa^*(G)\), and \(\kappa_f^*(G)\) are discussed. Moreover, some sufficient conditions for a graph to be \(\kappa_f^*\)-connected are presented. Furthermore, we introduce the concept of a spanning pipeline-connectivity and discuss some sufficient conditions for a graph to be \(k^*\)-pipeline-connected.