Abstract Link residual closeness is a newly proposed measure for network vulnerability. In this model, vertices are perfectly reliable and the links fail independently of each other. It measures the vulnerability even when the removal of links does not disconnect the graph. In this paper, we characterize those graphs that maximize the link residual closeness over the connected graphs with fixed order and one additional parameter such as connectivity, edge connectivity, bipartiteness, independence number, matching number, chromatic number, number of cut vertices and number of cut edges.