The degree-diameter problem for sparse graph classes

Abstract

The degree-diameter problem asks for the maximum number of vertices in agraph with maximum degree and diameter k. For xed k, the answer is (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 (k􀀀1), and for graphs of bounded arboricity the answer is (bk=2c), in both cases for xed 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