The decycling number of a graph G, denoted by \(\nabla (G)\) , is the smallest number of vertices whose removal results in an acyclic subgraph of G. A decycling set S of G with \(\nabla (G)\) vertices is said to be a \(\nabla \) -set. For any connected loopless 4-regular graph G on n vertices, it is shown that \( \frac{n+1}{3}\le \nabla (G)\le \frac{n+1}{2}\) , and presents a necessary and sufficient condition for the two equalities, respectively. Moreover, for any \(\nabla \) -set S of G, it is also shown that if \(G-S\) is a tree and \(\nabla (G)=\frac{n+1}{2}\) or \(\frac{n}{2}\) , then G is upper-embeddable. Meanwhile, there exists a Xuong-tree \(T_{X}\) of G such that vertices of S are leaves of \(T_{X}\) .
