The Spanning Fan-connectivity and the Spanning Pipeline-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 is w∗-connected for 1 ≤ w ≤ k if G is 1∗-connected. Let x be a vertex in G and let U = {y1, y2, . . . , yk} be a subset of V (G) where x is not in U . 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=1V (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 / ∈ U . The spanning fan-connectivity of a graph G, κf (G), is defined as the largest inte∗This work was supported in part by the National Science Council of the Republic of China under Contract NSC 94-2213-E-233-008. ger k such that G is w∗ f -connected for 1 ≤ w ≤ k if G is 1f -connected. In this paper, some relationship between κ(G), κ∗(G), and κf (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.