On the spanning fan-connectivity of graphs

AbstractLet 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=1kV(Pi)=V(G). A graph G is k∗-fan-connected (or kf∗-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 integer k such that G is wf∗-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 kf∗-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