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This paper proposes methods for reducing the maximum degree of vertices in graphs that maintain optimal broadcast time when a vertex can call a vertex at distance at most k during any time unit. The basic idea behind the proposed construction method is to eliminate edges from binary n-cubes. We show that, by this approach, the maximum degree of a vertex can be reduced to at most (2k - 1) ⌈ √log2|V| - k⌉, where 2 ≤ k 2 |V|, which asymptotically achieves the lower bound provided |V| = 2n and a constant k.