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 ∆ and the diameter D, was introduced in [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,∆ = 4 and k = 2,∆ = 3, we give constructions that result in sharper lower bounds.