Distributed Strong Diameter Network Decomposition

For a pair of positive parameters D,χ, a partition P of the vertex set V of an n-vertex graph G = (V,E) into disjoint clusters of diameter at most D each is called a (D,χ) network decomposition, if the supergraph G(P), obtained by contracting each of the clusters of P , can be properly χ-colored. The decompositionP is said to be strong (resp., weak) if each of the clusters has strong (resp., weak) diameter at most D, i.e., if for every cluster C ∈ P and every two vertices u, v ∈ C, the distance between them in the induced graph G(C) of C (resp., in G) is at most D. Network decomposition is a powerful construct, very useful in distributed computing and beyond. It was introduced by Awerbuch et. al. [AGLP89] in the end of the eighties. These authors showed that strong (2 √