Optimal Edge Fault-Tolerant Bijective Embedding of a Complete Graph over a Cycle

Abstract

An embedding of a guest graph G over a host graph H is an injective map Φ from the vertices of G to the vertices of H and a routing map ρ, which associates every edge e=xy in G to a Φ(x)-Φ(y) path ρ(e) in H. Given an edge f in H the number of edges e in G such that f belongs to ρ(e) is the (edge) congestion cong(f) of f. The length of ρ(e) is called the dilatation dil(e) of e. The sum of all th dilatations is the cost of the embedding. The removal of an edge f of H gives rise to a surviving graphGf, consisting of the guest graph without those edges that cross f, i.e., Gf=G−{e:f∈ρ(e)}. Given n and b, we are facing the problem of finding a minimum cost embedding Φ of a graph G with n vertices over the cycle Cn, such that for any surviving graph Gf, there is an embedding of the complete graph Kn over Gf whose congestions are not greater than b. This work presents a lower bound for the optimal cost of such problem and a family of embeddings that match this bound over a broad range of combinations of n and b.