maximum coverage version of dominating set
The problem in which you must select $k$ vertices to maximize the number of vertices dominated is known as the budgeted dominating set problem. The problem or its connected variant is studied at least by Lamprou, Sigalis and Zissimopoulos [1] and Khuller, Purohit and Sarpatwar [2]. It also appears in the recent survey of Narayanaswamy and Vijayaragunathan [3].



[1] Lamprou, Ioannis, Ioannis Sigalas, and Vassilis Zissimopoulos. "Improved Budgeted Connected Domination and Budgeted Edge-Vertex Domination." arXiv preprint arXiv:1907.06576 (2019).

[2] Khuller, Samir, Manish Purohit, and Kanthi K. Sarpatwar. "Analyzing the optimal neighborhood: Algorithms for budgeted and partial connected dominating set problems." Proceedings of the twenty-fifth annual ACM-SIAM Symposium on Discrete algorithms. Society for Industrial and Applied Mathematics, 2014.

[3] Narayanaswamy, N. S., and R. Vijayaragunathan. "Parameterized Optimization in Uncertain Graphs—A Survey and Some Results." Algorithms 13.1 (2020): 3.