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The first Zagreb index of a graph $G$ is the sum of the square of everyvertex degree, while the second Zagreb index is the sum of the product ofvertex degrees of each edge over all edges. In our work, we solve an openquestion about Zagreb indices of graphs with given number of cut vertices. Thesharp lower bounds are obtained for these indices of graphs in$\mathbb{V}_{n,k}$, where $\mathbb{V}_{n, k}$ denotes the set of all $n$-vertexgraphs with $k$ cut vertices and at least one cycle. As consequences, thosegraphs with the smallest Zagreb indices are characterized.

On the sharp lower bounds of Zagreb indices of graphs with given number of cut vertices