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A vertex set \( D \) of a graph \( G \) is a dominating set if every vertex not in \( D \) is adjacent to some vertex in \( D \). The domination number \( \gamma \) of a graph \( G \) is the minimum cardinality of a dominating set in \( G \). In 1989, Brigham and Dutton [1] proved
\[
\gamma(G) \leq \left\lceil\frac{3n-g}{6}\right\rceil
\]
for each graph \( G \) of order \( n \), minimum degree \( \delta \geq 2 \), and girth \( g \geq 5 \). For this class of graphs, Volkmann [8] recently gave the better bound
\[
\gamma(G) \leq \left\lceil\frac{3n-g-6}{8}\right\rceil
\]
if \( G \) is neither a cycle nor one of two exceptional graphs. If \( G \) is a graph of order \( n \), minimum degree \( \delta \geq 2 \), girth \( g \geq 5 \), then we show in this paper that
\[
\gamma(G) \leq \left\lceil\frac{3n-g-9}{6}\right\rceil
\]
if \( G \) is neither a cycle nor one of 40 exceptional graphs of order between 8 and 21.