On the spanning fan-connectivity of graphs

Let G be a graph. The connectivity of G, κ(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, κ^∗(G), is the maximum integer k such that G is w^∗-connected for 1≤w≤k if G is 1^∗-connected.Let x be a vertex in G and let U={y₁,y₂,…,y_k} be a subset of V(G) where x is not in U. A spanningk−(x,U)-fan, F_k(x,U), is a set of internally-disjoint paths {P₁,P₂,…,P_k} such that P_i is a path connecting x to y_i for 1≤i≤k and . A graph G is k^∗-fan-connected (or -connected) if there exists a spanning F_k(x,U)-fan for every choice of x and U with |U|=k and x∉U. The spanning fan-connectivity of a graph G, , is defined as the largest integer k such that G is -connected for 1≤w≤k if G is -connected.In this paper, some relationship between κ(G), κ^∗(G), and are discussed. Moreover, some sufficient conditions for a graph to be -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.