On the spanning w-wide diameter of the star graph

Let u and v be any two distinct nodes of an undirected graph G, which is k -connected. A container C(u,v ) between u and v is a set of internally disjoint paths {P1,P2, . . . ,Pw } between u and v where 1 ≤ w ≤ k . The width of C(u,v ) is w and the length of C(u,v ) {written as l [C(u,v )]} is max{l (Pi ) | 1 ≤ i ≤ w }. A w -container C(u,v ) is a container with width w . The w -wide distance between u and v , dw (u,v ), is min{l (C(u,v )) | C(u,v ) is a w -container}. A w -container C(u,v ) of the graph G is a w ∗-container if every node of G is incident with a path in C(u,v ). That means that the w -container C(u,v ) spans the whole graph. Let Sn be the n-dimensional star graph with n ≥ 5. It is known that Sn is bipartite. In this article, we show that, for any pair of distinct nodes u and v in different partite sets of Sn , there exists an (n − 1)∗-container C(u,v ) and the (n − 1)-wide distance d(n−1)(u,v ) is less than or equal to n! n−2 + 1. In addition, we also show the existence of a 2∗-container C(u,v ) and the 2-wide distance d2(u,v ) is bounded above by n! 2 + 1. © 2006 Wiley Periodicals, Inc. NETWORKS, Vol. 48(4), 235–249 2006