The Maximum Degree-and-Diameter-Bounded Subgraph in the Mesh

TL;DR

This paper investigates the largest connected subgraph within a mesh that adheres to specific degree and diameter constraints, providing bounds and constructions for particular cases relevant to parallel architectures.

Contribution

It extends the Degree-Diameter Problem to mesh graphs, offering new bounds and explicit constructions for certain dimensions and degree constraints.

Findings

Derived general bounds for arbitrary dimensions.

Provided sharper lower bounds for specific cases.

Constructed subgraphs achieving these bounds.

Abstract

The problem of finding the largest connected subgraph of a given undirected host graph, subject to constraints on the maximum degree $\Delta$ and the diameter $D$, was introduced in \cite{maxddbs}, 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 $k=3, \Delta = 4$ and $k=2, \Delta = 3$, we give constructions that result in sharper lower bounds.