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In 2022, the second author found a prolific construction of strongly regulargraphs, which is based on joining a coclique and a divisible design graph withcertain parameters. The construction produces strongly regular graphs with thesame parameters as the complement of the symplectic graph $\mathsf{Sp}(2d,q)$.  In this paper, we determine the parameters of strongly regular graphs whichadmit a decomposition into a divisible design graph and a coclique attainingthe Hoffman bound. In particular, it is shown that when the least eigenvalue ofsuch a strongly regular graph is a prime power, its parameters coincide withthose of the complement of $\mathsf{Sp}(2d,q)$. Furthermore, a generalizationof the construction is discussed.

Strongly regular graphs decomposable into a divisible design graph and a Hoffman coclique