d-separated paths in hypercubes and star graphs

In this paper, we consider a generalized node disjoint paths problem: d-separated paths problem. In a graph G, given two distinct nodes s and t, two paths P and Q, connecting s and t, are d-separated if d/sub G-{s,t}//spl ges/d for any u/spl isin/P-{s,t} and v/spl isin/Q-{s,t}, where d/sub G-{s,t}/(u,v) is the distance between u and v in the reduced graph G-{s,t}. d-separated paths problem is to find as many d-separated paths between s and t as possible. In this paper, we give the following results on d-separated paths problems on n-dimensional hypercubes H/sub n/ and star graphs G/sub n/. Given s and t in H/sub n/, there are at least (n-2) 2-separated paths between s and t. (n-2) is the maximum number of 2-separated paths between s and t for d(s,t)/spl ges/4. Moreover, (n-2) and separated paths of length at most d(s,t)+2 for d(s,t)