The Degree Diameter Problem for General Digraphs

Introduction
The degree/diameter problem for general digraphs can be stated as follows:

Given natural numbers d and k, find the largest possible number DN(d,k) of vertices in a digraph of maximum out-degree d and diameter k.

In attempting to settle the values of DN(d,k), research activities in this problem have follow the following two directions:


 * Increasing the lower bounds for DN(d,k) by constructing ever larger graphs.

whose order is close to the directed Moore bounds DM(d,k)=(dk+1-1)(d-2)-1.
 * Lowering and/or setting upper bounds for DN(d,k) by proving the non-existence of digraphs

Increasing the lower bounds for DN(d,k)
Considering constructions of large digraphs, the best results are obtained by Alegre digraph and its line digraphs, and by Kautz digraphs. Indeed, for maximum out-degree d=2 and diameter k&ge;4, the lower bounds 25&times;2k-4 are provided by the Alegre digraph and its line digraphs, while for the other combinations of d and k, the lower bounds dk+dk-1 are provided by Kautz digraphs.

Below is the table of the largest known digraphs (as of October 2008) in the degree diameter problem for digraphs of out-degree at most 3 ≤ d ≤ 16 and diameter 2 ≤ k ≤ 10. Only a few of the digraphs in this table are known to be optimal (marked in bold), and thus, finding a larger digraph whose order (in terms of the order of the vertex set) to the directed Moore bound is considered an open problem.

Table of the orders of the largest known digraphs for the directed degree diameter problem
Digraphs in bold are optimal.

The following table is the key to the colors in the table presented above:

Lowering and/or setting upper bounds for DN(d,k)
The directed Moore bound can be reached only for trivial combinations of d and k, that is, d=1 or k=1. For d=1 the Moore digraphs are the cycles on k+1 vertices (k&ge;1), while for k=1 they are the complete digraphs of order d+1. This outcome was obtained by Plesnik and Znam, and independently by Bridges and Toueg. Therefore, theoretical works have been done to determine the lowest possible upper bounds. In this direction reserachers have been interested in digraphs of maximum out-degree d, diameter k and order DM(d,k)-&delta; for small &delta;. The parameter &delta; is called the defect. Such digraphs are called (d,k,-&delta;)-digraphs.

This research direction represents a very open reseach area.

For &delta;=1 Gimbert proved that line digraphs of complete digraphs are the only (d,2,-1)-digraphs for d&ge;3. Miller and Fris showed the non-existence of (2,k,-1)-digraphs for k&ge;3, while Baskoro, Miller, Širáň and Sutton proved the non-existence of (3,k,-1)-digraphs for k&ge;3. Conde, Gimbert, Gonzalez, Miret and Moreno proved the non-existence of (d,3,-1)-digraphs for d&ge;3.

When &delta;=2 Miller and Širáň proved that there exist no (2,k,-2)-digraphs for k&ge;2.

For any other combination of d, k and &delta;, the problem of deciding the non-existence or otherwise of (d,k,-&delta;)-digraphs remains open.