On the t-th level cop number

In the police department, the police chief decides ending up the crimes before a deadline by not letting the robber commit damages. She follows a discipline to use the least number of officers. This is our motivation to define a new concept [Formula: see text]-th level cop number [Formula: see text] for the classical game Cops and Robber, which is played on the underlying graph [Formula: see text]. It has a close connection with [Formula: see text]-distance domination number of [Formula: see text]. A lower bound for [Formula: see text] is presented by [Formula: see text]-distance domination number of [Formula: see text]. For some special cases, we observe that [Formula: see text] and [Formula: see text]-distance domination number are equal.