Let ∇(G) and γMfalse(Gfalse) denote the minimum size of a decycling set and maximum genus of a graph G, respectively. For a connected cubic graph G of order n, it is shown that ∇false(Gfalse)+γMfalse(Gfalse)=n2+1. Applying the formula, we obtain some new results on the decycling number and maximum genus of cubic graphs. Furthermore, it is shown that the number of vertices of a decycling set S in a k‐regular graph G is false|Sfalse|=1k−1false{β(G)+c+mS′−1false}, where c and mS′ are the number of components of G−S and the number of edges in G[S], respectively. Therefore, S is minimum if and only if c+mS′−1 is minimum. As an application, this leads to a lower bound ∇false(Gfalse)≥false⌈βfalse(Gfalse)k−1false⌉ for ∇(G) of a k‐regular graph G. In many cases this bound may be sharp.

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