Shortest Paths in Sierpiński Graphs

In [23], Klavzar and Milutinovic (1997) proved that there exist at most two different shortest paths between any two vertices in Sierpinski graphs S-k(n), and showed that the number of shortest paths between any fixed pair of vertices of S-k(n) can be computed in O(n). An almost-extreme vertex of S-k(n), which was introduced in Klavzar and Zemljic (2013) [27], is a vertex that is either adjacent to an extreme vertex or incident to an edge between two subgraphs of S-k(n) isomorphic to S-k(n-1). In this paper, we completely determine the set S-u = {v is an element of V(S-k(n)) : there exist two shortest u, v-paths in S-k(n)}, where u is any almost-extreme vertex of S-k(n). (C) 2013 Elsevier B.V. All rights reserved.