The maximum degree and diameter-bounded subgraph in the mesh

The problem of finding the largest connected subgraph of a given undirected host graph, subject to constraints on the maximum degree Δ and the diameter D, was introduced in Dekker et al. (2012) [1], as a generalization of the Degree–Diameter Problem. A case of special interest is when the host graph is a common parallel architecture. Here we discuss the case when the host graph is a k-dimensional mesh. We provide some general bounds for the order of the largest subgraph in arbitrary dimension k, and for the particular cases of k=3,Δ=4 and k=2,Δ=3, we give constructions that result in sharper lower bounds.