Sharper Lower Bounds in the Maximum Degree and Diameter Bounded Subgraph Problem in the Mesh

TL;DR

This paper establishes improved lower bounds for the maximum size of subgraphs within k-dimensional meshes that have bounded degree and diameter, advancing understanding of the MaxDDBS problem in these structures.

Contribution

It provides new lower bounds for MaxDDBS in k-dimensional meshes that closely match known upper bounds, especially for  and  degrees, improving theoretical understanding.

Findings

Lower bounds match upper bounds up to first two terms for  and higher.

For , lower bounds are of the same order as upper bounds.

Results hold for all dimensions k and degrees .

Abstract

The Maximum Degree and Diameter Bounded Subgraph Problem (MaxDDBS) asks: given a host graph G, a bound on maximum degree \Delta, and a diameter D, what is the largest subgraph of the host graph with degree bounded by \Delta and diameter bounded by D? In this paper, we investigate this problem when the host graph is the k-dimensional mesh. We provide lower bounds for the size of the largest subgraph of the mesh satisfying MaxDDBS for all k and \Delta > 3 that agree with the known upper bounds up to the first two terms, and show that for \Delta = 3, the lower bounds are at least the same order of growth as the upper bounds.