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, Ck(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 isw-connected for 1 ≤ w ≤ k if G is 1-connected. Let x be a vertex inG and letU = {y1, y2, . . . , yk} be a subset of V (G)where x is not inU . A spanning k− (x,U)-fan, Fk(x,U), is a set of internally-disjoint paths {P1, P2, . . . , Pk} such that Pi is a path connecting x to yi for 1 ≤ i ≤ k and∪i=1 V (Pi) = V (G). A graph G is k -fanconnected (or k∗f -connected) if there exists a spanning Fk(x,U)-fan for every choice of x and U with |U| = k and x 6∈ U . The spanning fan-connectivity of a graph G, κ f (G), is defined as the largest integer k such that G isw f -connected for 1 ≤ w ≤ k if G is 1 ∗ f -connected. In this paper, some relationship between κ(G), κ(G), and κ f (G) are discussed. Moreover, some sufficient conditions for a graph to be k∗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. Published by Elsevier B.V.