Fault-tolerant embedding of starlike trees into restricted hypercube-like graphs

Abstract

A d-starlike tree (or a d-quasistar) is a subdivision of a star tree of degree d. A family of hypercube-like interconnection networks, called restricted hypercube-like graphs, includes most non-bipartite hypercube-like networks found in the literature such as twisted cubes, crossed cubes, Möbius cubes, recursive circulant G(2m,4) of odd m, etc. In this paper, we prove that given an arbitrary fault-free vertex r in an m-dimensional restricted hypercube-like graph with a set F of faults (vertex and/or edge faults) and d positive integers, l1,l2,…,ld, whose sum is equal to the number of fault-free vertices minus one, there exists a d-starlike tree rooted at r, each of whose subtrees forms a fault-free path on li vertices for i∈{1,2,…,d}, provided |F|≤m−2 and |F|+d≤m. The bounds on |F| and |F|+d are the maximum possible.