Here, imagine the entire network as a graph, computers as nodes and links as edges. They are connected by wires ie it forms connected component.
So now, you have to find the time complexity of the largest connected component whose removal may not affect the connectivity of the graph.
The largest connected component can be found by DFS traversal on the graph whose time complexity is O(V+E) using adjacency list representation.
So, using this you can find out the time complexity.