The Degree-Diameter Problem for Sparse Graph Classes

The degree-diameter problem asks for the maximum number of vertices in a graph with maximum degree ∆ and diameter k. For fixed k, the answer is Θ(∆k). We consider the degree-diameter problem for particular classes of sparse graphs, and establish the following results. For graphs of bounded average degree the answer is Θ(∆k−1), and for graphs of bounded arboricity the answer is Θ(∆bk/2c), in both cases for fixed k. For graphs of given treewidth, we determine the the maximum number of vertices up to a constant factor. Other precise bounds are given for graphs embeddable on a given surface and apex-minor-free graphs.