On the spanning fan-connectivity of graphs

• Authors: Cheng-Kuan Lin;Jimmy J. M. Tan;D. Frank Hsu;Lih-Hsing Hsu
• Venue: Discrete Applied Mathematics
• Year: 2009

Let G be a graph. The connectivity of G, @k(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, @k^*(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={y"1,y"2,...,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 {P"1,P"2,...,P"k} such that P"i is a path connecting x to y"i for 1@?i@?k and @?"i"="1^kV(P"i)=V(G). A graph G is k^*-fan-connected (or k"f^*-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, @k"f^*(G), is defined as the largest integer k such that G is w"f^*-connected for 1@?w@?k if G is 1"f^*-connected. In this paper, some relationship between @k(G), @k^*(G), and @k"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.