A strongly regular decomposition of a strongly regular graph is a partition of the vertex set into two parts on which the induced subgraphs are strongly regular, or cliques or cocliques. Strongly regular designs (srd's) as defined by D. G. Higman are coherent configurations of rank 10 with two fibers in which the homogeneous configuration on each fiber is a strongly regular graph. Haemers and Higman proved the equivalence between strongly regular decompositions, excluding special cases, and srd's with certain parameter conditions. Here we obtain this result by examining the srd's that admit a fusion to a strongly regular graph on the full vertex set. We derive equivalent conditions to Theorem 2.8 of Haemers and Higman by elementary methods. Incorporating recent works of Hanaki and Klin and Reichard, a table of feasible parameter sets for this class of srd's is presented along with a discussion of known constructions. In two cases, nonexistence is observed due to nonexistence of the strongly regular graph obtained through fusion. One of these is also ruled out by Hobart's generalized Krein conditions, applied to srd's. As strongly regular decompositions of the complete graph have sparked interest with recent papers we observe that in our situation this occurs only when the constituent graphs are also complete and the design is trivial.

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关于允许融合到强正则分解的强正则设计

强正则图的强正则分解是将顶点集划分为两个部分，在这两个部分上，诱导子图是强正则的，或者是团或团。DG Higman 定义的强规则设计 (srd) 是具有两条纤维的 10 级相干配置，其中每条纤维上的均匀配置是一个强规则图。Haemers 和 Higman 证明了强正则分解（不包括特殊情况）与具有某些参数条件的 srd 分解之间的等价性。在这里，我们通过检查允许融合到全顶点集上的强正则图的 srd 来获得这个结果。我们通过基本方法推导出与 Haemers 和 Higman 定理 2.8 等效的条件。结合 Hanaki 和 Klin 以及 Reichard 的近期作品，提供了此类 srd 的可行参数集表以及对已知结构的讨论。在两种情况下，由于通过融合获得的强规则图不存在，因此观察到不存在。其中之一也被适用于 srd 的 Hobart 的广义 Kerin 条件排除。由于完整图的强规则分解引起了最近论文的兴趣，我们观察到，在我们的情况下，只有在组成图也完整且设计微不足道时才会发生这种情况。