In view of some shortcomings of traditional vertex 1-center (V1C), we introduce a vertex quickest 1-center (VQ1C) problem on a tree, which aims to find a vertex such that the maximum transmission time to transmit Formula omitted units data is minimum. We first characterize some intrinsic properties of VQ1C and design a binary search algorithm in Formula omitted time based on the relationship between V1C and VQ1C, where n is the number of vertices. Furthermore, we investigate the inverse VQ1C problem under weighted Formula omitted norm, in which we modify a given capacity vector in an optimal way such that a prespecified vertex becomes the vertex quickest 1-center. We introduce a concept of an effective modification and provide some optimality conditions for the problem. Then we propose an Formula omitted time algorithm. Finally, we show some numerical experiments to verify the efficiency of the algorithms.