A \((k,g)\)-graph is a k-regular graph of girth \(g\). Given \(k\ge 2\) and \(g\ge 3\), \((k,g)\)-graphs of infinitely many orders are known to exist and the problem of finding a (k, g)-graph of the smallest possible order is known as the Cage Problem. The aim of our paper is to develop systematic (programmable) ways for lowering the orders of existing \((k,g)\)-graphs, while preserving their regularity and girth. Such methods, in analogy with the previously used excision, may have the potential for constructing smaller (k, g)-graphs from current smallest examples—record holders—some of which have not been improved in years. In addition, we consider constructions that preserve the regularity, the girth and the order of the considered graphs, but alter the graphs enough to possibly make them suitable for the application of our order decreasing methods. We include a detailed discussion of several specific parameter cases for which several non-isomorphic smallest examples are known to exist, and address the question of the distance between these non-isomorphic examples based on the number of changes required to move from one example to another.

\ ((k,g)\)图是周长\(g\)的k正则图。给定\(k\ge 2\)和\(g\ge 3\)，\((k,g)\) -已知存在无限多阶图，并且存在找到 ( k ,  g )-的问题最小可能阶数的图被称为笼问题。我们论文的目的是开发系统（可编程）方法来降低现有\((k,g)\)图的阶数，同时保留其规律性和周长。与之前使用的切除类似，此类方法可能有潜力从当前最小的示例（记录保持者）构造更小的 ( k ,  g ) 图，其中一些示例多年来一直没有得到改进。此外，我们考虑保留所考虑的图的规律性、周长和顺序的结构，但对图进行足够的改变以使其适合我们的降序方法的应用。我们详细讨论了已知存在几个非同构最小示例的几种特定参数情况，并根据从一个示例移动到另一个示例所需的更改数量来解决这些非同构示例之间的距离问题。